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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction .
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so.. what would 6 ( g+3 ) be ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ?
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what is 4 ( 6-8*9+5 ) +-5 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ?
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i cut off a piece of length 4 '' what is the area of the remaining piece ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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when will you ever need to use expressions in life ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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what is the difference between the law of multiplication and the addition rule ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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so we only use the distribution property when we have things in brackets ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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when do i know to keep the brackets and when not too ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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when do we decide to put/or have to put parenthesis and when not to ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is and how do you do distributive property of division ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is a distributed property/ law ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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so , how would you answer a problem say its like 3 loaves of bread for $ 1.99 how would you awnser that with the distributive property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression .
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does sal use the distributive law 7 ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 .
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so what exactly does distribute mean ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how would the distributive property work if the problem is ( 3-6 ) ^2 ( where the two is to the second power of ( 3-6 ) ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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what is the commutative property ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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can any one tell me how to apply the distributive property to everyday life or who uses this type of math ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ?
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why did sal say to add all the blue and red circles up when you could multiply them ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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how do you use the distributive property using variables and numbers ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works .
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why is the distributive property so useful ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles .
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if you had an negative number would the negative be distributed to the equation inside the bracket or not ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses .
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also if you had two numbers in front of the bracket and the first was a positive and the second was an negative would both numbers be distributed or not ?
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression . so let 's just try to solve this or evaluate this expression , then we 'll talk a little bit about the distributive law of multiplication over addition , usually just called the distributive law . so we have 4 times 8 plus 8 plus 3 . now there 's two ways to do it . normally , when you have parentheses , your inclination is , well , let me just evaluate what 's in the parentheses first and then worry about what 's outside of the parentheses , and we can do that fairly easily here . we can evaluate what 8 plus 3 is . 8 plus 3 is 11 . so if we do that -- let me do that in this direction . so if we do that , we get 4 times , and in parentheses we have an 11 . 8 plus 3 is 11 , and then this is going to be equal to -- well , 4 times 11 is just 44 , so you can evaluate it that way . but they want us to use the distributive law of multiplication . we did not use the distributive law just now . we just evaluated the expression . we used the parentheses first , then multiplied by 4 . in the distributive law , we multiply by 4 first . and it 's called the distributive law because you distribute the 4 , and we 're going to think about what that means . so in the distributive law , what this will become , it 'll become 4 times 8 plus 4 times 3 , and we 're going to think about why that is in a second . so this is going to be equal to 4 times 8 plus 4 times 3 . a lot of people 's first instinct is just to multiply the 4 times the 8 , but no ! you have to distribute the 4 . you have to multiply it times the 8 and times the 3 . this is right here . this is the distributive property in action right here . distributive property in action . and then when you evaluate it -- and i 'm going to show you in kind of a visual way why this works . but then when you evaluate it , 4 times 8 -- i 'll do this in a different color -- 4 times 8 is 32 , and then so we have 32 plus 4 times 3 . 4 times 3 is 12 and 32 plus 12 is equal to 44 . that is also equal to 44 , so you can get it either way . but when they want us to use the distributive law , you 'd distribute the 4 first . now let 's think about why that happens . let 's visualize just what 8 plus 3 is . let me draw eight of something . so one , two , three , four , five , six , seven , eight , right ? and then we 're going to add to that three of something , of maybe the same thing . one , two , three . so you can imagine this is what we have inside of the parentheses . we have 8 circles plus 3 circles . now , when we 're multiplying this whole thing , this whole thing times 4 , what does that mean ? well , that means we 're just going to add this to itself four times . let me do that with a copy and paste . copy and paste . let me copy and then let me paste . there you go . that 's two . that 's one , two , three , and then we have four , and we 're going to add them all together . so this is literally what ? four times , right ? let me go back to the drawing tool . we have it one , two , three , four times this expression , which is 8 plus 3 . now , what is this thing over here ? if you were to count all of this stuff , you would get 44 . but what is this thing over here ? well , that 's 8 added to itself four times . you could imagine you 're adding all of these . so what 's 8 added to itself four times ? that is 4 times 8 . so this is 4 times 8 , and what is this over here in the orange ? we have one , two , three , four times . well , each time we have three . so it 's 4 times this right here . this right here is 4 times 3 . so you see why the distributive property works . if you do 4 times 8 plus 3 , you have to multiply -- when you , i guess you could imagine , duplicate the thing four times , both the 8 and the 3 is getting duplicated four times or it 's being added to itself four times , and that 's why we distribute the 4 .
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rewrite the expression 4 times , and then in parentheses we have 8 plus 3 , using the distributive law of multiplication over addition . then simplify the expression .
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does butt chicken tastes good ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level .
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what is the difference between a `` estimated '' number and a `` projected '' number ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations .
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i 'm just thinking how bad this will be in 30 years from now when i retire , will the state cut/will i lose my promised pension altogether or , as selfish as it sounds , will the state still pay up at the cost of cutting major funding for other things like they have been doing ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions .
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do n't you think a numbers guy should shop around for a better rate ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest .
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does illinois report its funding ratio assuming the more reasonable rates of return expressed in the previous video and if so , what is its funded status at those rates of return ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside .
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why should the state fund pensions ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % .
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is n't it the company 's fault for having underfunded pensions ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % .
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can illinois get out of dept with out raising taxes ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded .
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when the inflation rate start to increase past 3 percent , would the cola automatically increase to match the new higher inflation rate ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside .
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my question is , are any of the state pension funds managed by our `` too big to fail '' brokerage firms , like j.p. morgan chase ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois .
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if we forced them all to take bailout money , why ca n't we ask them to responsibly manage pensions ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level .
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does this mean defined benefit pension plans will become dramatically underfunded all over again - at all levels of government including federal ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ?
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what is ma 's under funding percentage ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically .
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we are always in a low rate but why ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy .
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why does your discussion assume that the only way a state can meet its pension obligations is by giving money to the private sector to invest ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest .
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what would be wrong with paying-as-you-go : that is , paying each year 's obligations to retirees in the years in which they are entitled to pensions , instead of entering the weird world of predictions of ( a ) future benefit starting amounts , ( b ) future inflations rates , ( c ) future rates of investment return ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states .
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what is the recommended level of funding by defined benefit plans ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation .
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also does n't the funding level get tricky with the return on investments being harder to project ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy .
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was the illinois pension fund invested in less safe investments than most pension funds and thus lost a lot of money during the 2008 financial crisis ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ?
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what is massachusetts u.f ( under funding ) percentage ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically .
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how much of the 2010 pension shortfall is artificially low because it is reflecting the low market value of stocks , funds , and other investments in 2010 because of the financial crisis in 2008 ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion .
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is the situation in ct similar to that described for illinois ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states .
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why is puerto rico not included in the map ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million .
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what are the benifits to not paying your prompised liabilities ?
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in the last video , we talked about pensions and how they 're defined benefit plans and how they could to get underfunded or how there could be temptation for people to underfund them . in this video , i want to make things a little bit more concrete by looking at actual numbers , especially at the state level . so right over here is a map of , obviously , the united states . and what it shows is how funded the pension liabilities are in the different states . so for example -- actually , texas , for example , 83 % of their pension liabilities are funded . they 've set aside 83 % of the right amount of money to fund their pension obligations , not 100 % . it is underfunded , but it 's not crazy . california , pretty high , 78 % . but one of these states is probably jumping out at you , probably because it has been shaded in red . and that is the state of illinois , and illinois is in trouble because it 's only funded 45 % of its pension obligations . and illinois really jumps out because it 's in red , but there 's a lot of states that are pretty close to illinois . louisiana , 56 % . oklahoma , 56 % . kentucky , 54 % . west virginia , 58 % . and this is an issue because they 've set aside , in the past , very little money for the pension obligations that are starting to hit now , especially that you have a retiring baby boomer population . and in order to meet those obligations , those promised obligations , they 're going to have to dig into money that was being spent other places , that going in the past when they were underfunding the pension , they were able to fund other things nicely , but not fund the pension and kind of kick the can down the road . but now that the can ca n't be kicked any further , it 's going to have to go the other way around . you 're going to have to take money from other things to fund your pensions . and to make it clear , let 's focus on the state of illinois . so this right over here . there 's a couple of things going on . in this kind of yellow ochre color -- and i 'll circle it in yellow ochre -- they were talking about the total liabilities . and just to make this graph clear , it 's not just the yellow ochre part that 's total liabilities . the entire height of each of these bars is the total liabilities , and you see how it has just completely blossomed here . and there 's a lot of things that go into the total liabilities , the same things that we talked about in the last video . there are things like return on investment . if you are in a low interest rate environment , like we are now -- for example , my money in my savings account , i think , is getting like 0.4 % interest . it 's getting pretty much no interest . if you 're in a low interest rate environment , if you 're not getting good returns -- and a lot of pensions tend to go into very safe assets , but those are getting very low returns . you 're going to have to set aside more money , and so you see these obligations essentially just growing dramatically . on top of that , you have things like cost of living adjustments . these are attempts at kind of factoring in inflation , how much things are costing in that region . but they are also sometimes negotiated . and sometimes , and especially in the case of illinois , they 've grown faster than the rate of inflation . and so you have these liabilities , and you see that they 're getting less and less well funded . so if we go right over here , this is what this green line is , the funding ratio . so how well funded are these liabilities ? say the red part of the bar is the part that is not paid for . and the green is the ratio of the red or is the ratio of what is funded , essentially this higher part . it 's the ratio of this part right over here to the entire bar . and you see right over here , illinois is in a bad situation . their total liabilities are 138 billion . this is in millions , so it 's 138,000 million . so it 's 138 billion . this is for one state . and 85 or 86 billion of that is unfunded , that they have to figure out some way to get the money because the right amount of money was not being set aside . and to do that , they 're going to have to dig in into other things . so this right over here , this is the pension contribution . let me circle this . so in this yellow color , once again , this is the pension contribution . and now the state , they 're going to have to -- in order to get to a funded position , they 're going to make up for all of the underfunding of the past and also the other factors that are making this obligation even larger . they 're going to have to dig into other things . so you see right over here in yellow , these are the contributions that they 're going to have to make for the pension . and you see that growing . it 's growing to in excess by 2018 of $ 6 billion a year . but what 's really fascinating about this graph is it 's passing up total education funding in the state . so the cost of funding retirements for people who have already done service for the state but are n't in service to the state right now is going to pass up -- and this is happening very soon -- is going to pass up actual spending on a state-wide basis on education . and at the state level , education is a major , major , major expenditure . so it 's going to be passing up a major expenditure , very important expenditure for the future of the state based on past obligations . and to understand where this is going -- and just to understand illinois ' situation , there 's 750,000 illinois -- i do n't know how to say this -- illinoians , illinoisians , who are members of the state 's five pension system . so this is the teachers retirement system . this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that . there 's many more teachers who 've been retired , many who have been in the state universities . and this right over here is the general assembly . very few people who are in the illinois state assembly . and you can see kind of comparable salaries . this is the retirement benefit , not just the salaries . this is how much on average these folks are getting once they retire on an annual basis . so you see that they 're pretty reasonable , especially for the judges , although they are a small fraction . but in all fairness , this was promised to these people . they planned . the probably took lower compensation while they were working with the expectation that they would be able to get these benefits once they retired . they also probably stayed in the jobs longer . this is a way of retaining employees , because they knew that they were going to get this benefit . so you might say , oh , these are really , really great benefits . but at the same time , these people probably sacrificed other things in order to get these benefits now . but it 's a very hard question . when you look at this , you say , well , these people , they 've done service . they put these expectations . but at the same time , you 're like , well , this is really cutting in -- and this is just one thing that i 'm showing . it 's really cutting into very important areas of investment for the entire state . so the whole reason of really just surfacing this , this whole pension issue , is just to put this in . and hopefully people understand what the issues are , because that 's the only way that fairly hard decisions are going to have to be made , decisions on cutting necessary investment or restructuring or who knows what it might be . i do n't envy the people who have to make these decisions .
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this is the state universities retirement system . this is the state employees retirement system . this is the judges retirement system . you see there 's a lot fewer judges than that .
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why do governments utilize a different accounting system than private businesses ?
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f of x , then f of g of x is going to be equal to , well , everywhere you see an x , you would replace with g of x . so , it would be equal to the square root of g of x , g of x plus one , and g of x is x squared minus five . so it would be the square root of x squared minus five . that 's g of x , that 's g of x right over there , plus one , plus one . and this would be the square root of x squared minus four , x squared minus four . well , that 's not what they have here . they have x squared plus four . so this is n't going to be right . we could try this one . f of , i 'll do this in green just for fun . f of g of x is going to be equal to , it 's going to be the square root of , well everywhere you see an x , replace it with a g of x . the square root of g of x plus one , square root of g of x plus one , well , that 's gon na be equal to the square root , g of x is x squared minus one . so , g of x is x squared , oh wait , let me be careful here . this is going to be g of x plus nine , g of x plus nine , g of x plus nine , very important detail . we have a nine here , it 's gon na be a nine here . and this is going to be equal to the square root of g of x , is x squared minus five , x square minus five . and then you 're gon na add nine . you 're gon na add nine . well , what does that give us ? this gives us the square root of x squared minus five plus nine . that 's gon na be plus four . which is exactly what you have right over here . so , if f of x is equal to the square root of x plus nine , then f of g of x is going to be this . so , that 's our choice , right over there .
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ?
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what is a domain and range ?
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f of x , then f of g of x is going to be equal to , well , everywhere you see an x , you would replace with g of x . so , it would be equal to the square root of g of x , g of x plus one , and g of x is x squared minus five . so it would be the square root of x squared minus five . that 's g of x , that 's g of x right over there , plus one , plus one . and this would be the square root of x squared minus four , x squared minus four . well , that 's not what they have here . they have x squared plus four . so this is n't going to be right . we could try this one . f of , i 'll do this in green just for fun . f of g of x is going to be equal to , it 's going to be the square root of , well everywhere you see an x , replace it with a g of x . the square root of g of x plus one , square root of g of x plus one , well , that 's gon na be equal to the square root , g of x is x squared minus one . so , g of x is x squared , oh wait , let me be careful here . this is going to be g of x plus nine , g of x plus nine , g of x plus nine , very important detail . we have a nine here , it 's gon na be a nine here . and this is going to be equal to the square root of g of x , is x squared minus five , x square minus five . and then you 're gon na add nine . you 're gon na add nine . well , what does that give us ? this gives us the square root of x squared minus five plus nine . that 's gon na be plus four . which is exactly what you have right over here . so , if f of x is equal to the square root of x plus nine , then f of g of x is going to be this . so , that 's our choice , right over there .
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ?
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hi , my question is , how is x equal to g ( x ) ?
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f of x , then f of g of x is going to be equal to , well , everywhere you see an x , you would replace with g of x . so , it would be equal to the square root of g of x , g of x plus one , and g of x is x squared minus five . so it would be the square root of x squared minus five . that 's g of x , that 's g of x right over there , plus one , plus one . and this would be the square root of x squared minus four , x squared minus four . well , that 's not what they have here . they have x squared plus four . so this is n't going to be right . we could try this one . f of , i 'll do this in green just for fun . f of g of x is going to be equal to , it 's going to be the square root of , well everywhere you see an x , replace it with a g of x . the square root of g of x plus one , square root of g of x plus one , well , that 's gon na be equal to the square root , g of x is x squared minus one . so , g of x is x squared , oh wait , let me be careful here . this is going to be g of x plus nine , g of x plus nine , g of x plus nine , very important detail . we have a nine here , it 's gon na be a nine here . and this is going to be equal to the square root of g of x , is x squared minus five , x square minus five . and then you 're gon na add nine . you 're gon na add nine . well , what does that give us ? this gives us the square root of x squared minus five plus nine . that 's gon na be plus four . which is exactly what you have right over here . so , if f of x is equal to the square root of x plus nine , then f of g of x is going to be this . so , that 's our choice , right over there .
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ?
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i understand how you solve a function using operations , but how do you interpret the graph of a function ?
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ? all right , this is interesting . so , one way i could think about doing this is , well let 's just try each of these f of xes . if this is , if this was f of x , then f of g of x is going to be equal to , well , everywhere you see an x , you would replace with g of x . so , it would be equal to the square root of g of x , g of x plus one , and g of x is x squared minus five . so it would be the square root of x squared minus five . that 's g of x , that 's g of x right over there , plus one , plus one . and this would be the square root of x squared minus four , x squared minus four . well , that 's not what they have here . they have x squared plus four . so this is n't going to be right . we could try this one . f of , i 'll do this in green just for fun . f of g of x is going to be equal to , it 's going to be the square root of , well everywhere you see an x , replace it with a g of x . the square root of g of x plus one , square root of g of x plus one , well , that 's gon na be equal to the square root , g of x is x squared minus one . so , g of x is x squared , oh wait , let me be careful here . this is going to be g of x plus nine , g of x plus nine , g of x plus nine , very important detail . we have a nine here , it 's gon na be a nine here . and this is going to be equal to the square root of g of x , is x squared minus five , x square minus five . and then you 're gon na add nine . you 're gon na add nine . well , what does that give us ? this gives us the square root of x squared minus five plus nine . that 's gon na be plus four . which is exactly what you have right over here . so , if f of x is equal to the square root of x plus nine , then f of g of x is going to be this . so , that 's our choice , right over there .
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let g of x , let g of x equal x squared minus five . if f of g of x is equal to the square root of x squared plus four , which of the following describes f of x ?
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how do you know what the value of a function is by looking at its graph ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on .
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was the intuition of the mathematicians who defined matrix addition really that arbitrary ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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can matrices be considered as vectors , in a way , with the addition and subtraction properties of matrices being similar to that of vectors ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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what are matrices used for ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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what is the point of a matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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could there ever be a situation when you have to add or subtract matrices of different dimensions ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition .
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what are the point of matrixes ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five .
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i understand why you would n't be able to put the matrices in any order while dividing , but since multiplying is simply repeated addition , would n't the order of two matrices not matter ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two .
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can we divide a matrice with one other ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix .
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if i want to add two matrices with dimensions 3 x 2 and 2 x 3 , can i transpose one of them to get two matrices with dimensions 3 x 2 and 3 x 2 and then perform addition ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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or is it that such kinds of matrices with different dimensions can never be added ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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so if adding and subtracting different dimensional matrices is undefined , can you multiply and divide ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two .
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why ca n't you just put a zero in all the missing places and then add all the others ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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can we use properties such as the associative property or the distributive property while working with matrices ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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how do find the additive inverse of matrices ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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to add or subtract matrices of different dimensions , why would n't it work to fill in zeros in the matrices ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven .
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why is there a `` d '' after the equals sign ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined .
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is it just khan making a smiley face or is it used to indicate the sat choice `` d. undefined '' ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix .
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why ca n't we add zero as a place-holder for any missing rows/columns ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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how do you know if a matrix is unsolvable ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication .
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does the size of a and b have to be the same in order to do these operations ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven .
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for instance , could n't we rewrite the 2x2 matrix [ 5 7 ; -1 6 ] as a 3x2 [ 5 7 ; -1 6 ; 0 0 ] ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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if the addition and subtraction of different dimension matrices is undefined , is the same true for multiplication ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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is it possible to multiply and divide matrices ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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when dealing with arrays of different lengths ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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is it possible to divide or square root a matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix .
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can there be a third variable of some sort in a matrix besides rows and columns ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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how do you enter matrices into the computer ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five .
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do two matrixes have to have the same dimensions to be added ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . )
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waht would be the sum of 3+ [ a matrix ] ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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how do you do scalar matrix addition and subtraction ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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why is it then not possible to add two matrices with different dimensions in this aggregate way ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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how do you use matrices to solve nonmatrix problems in real life ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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for matrices with different dimensions , for instance a 2x3 matrix and 2x2 matrix , why ca n't we consider the latter to have a third column with all entries equal to zero so that we can add the two matrices ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven .
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would you change pi to 3.14 and add ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero .
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what if i wanted to add a 3x4 matrix to a 12x1 matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven .
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would [ 6 7 8 ] - [ 5 ] be undefined ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough .
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while multiplying two numbers with both negative signs , will it turn positive in the product part or negative ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 .
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is the final result of adding/subracting matrices always going to have only 1 column ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on .
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what does the word `` arbitrary '' mean ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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`` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix .
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so you ca n't add matrices of different dimensions ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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is it possible for a matrix to have a complex number in it ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two .
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at the last part of the video , could n't you just expand one of the matrices by writing in 0 's ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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could n't the dimensions be added or subtracted the same way as the numbers within the matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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are there any operations that can reduce or increase the initial size of an matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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is matrix addition is possible when there is the number of rows or columns are different ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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how do you multiply a number by a matrix ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on .
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what is an arbitrary number ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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i still ca n't figure out matrices where a matrix + b matrix = c matrix and it 's asking me to solve for x and y ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions .
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how do you solve matrices with variables ?
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let 's think about how we can define `` matrix addition . '' and mathematicians could have chosen any of an arbitrary number of ways to define addition . but they 've picked a way to define addition that seems – one – to make sense , and it also has nice properties that allow us to do interesting things with matrices later on . so if you were one of these mathematicians who were first defining how matrices should be added , how would you define adding this first matrix over here to the second one ? well , the most common-sense thing that might have jumped out at you – especially because these two matrices have the same dimensions – ( this is a 2-by-3 matrix . it has 2 rows and 3 columns . this is also a 2-by-3 matrix . it also has 2 rows and 3 columns . ) – is to just add the corresponding entries . and if that was your intuition , then you had the same intuition as the mathematical mainstream . that the addition of matrices should literally just be adding the corresponding entries . so in this situation , we would add 1 + 5 to get the corresponding entry in the sum – which is 6 . you can add negative seven plus zero to get negative seven . you can add five plus three to get eight . you can add -and i 'm running out of colours here- you could add zero plus eleven to get eleven . you can add three to negative one to get two . and you could add -and you could add negative ten plus seven to get negative three . and if you see this definition of matrix addition you see that it actually does not matter in what order that we actually add these matrices . i could 've done this the other way around , if i did this the other way around -so let me copy and paste this- so if i were to add this matrix -so let me copy , let me paste it- if i were to add that matrix to -let me copy and paste the other one- this matrix , copy and paste , you 'll see that the order in which i 'm adding the matrices does not matter so this is just like adding numbers . a plus b is just the same thing as b plus a . what we 'll see is this wo n't be true for every matrix operation that we study and in particular this will not be true for matrix multiplication . but if you add these two things , using the definition we just came up with , adding corresponding terms , you 'll get the exact same result . up here we added one plus five and we got six her we 'll add five plus one and we 'll get six . we get the same result because one plus five is the same thing as five plus one . here we have zero plus negative seven you get negative seven . so you 're going to get the exact same thing as we got up here . so when you 're adding matrices , if you were to call -if you were to call this matrix right over here matrix a which we normally denote with a capital , bolder letter , and you call this matrix right over here matrix b then when we take the sum of a plus b which is this thing right over here , and we see it 's the exact same thing as b , as matrix b plus matrix a . now let me ask you an interesting question . what if i wanted to subtract matrices ? so let 's once again think about matrices that have the same dimensions . so let 's say i 'm gon na do then two two-by-two matrices . so let 's say it 's zero , one , three , two , and from that i want to subtract negative one , three , zero , and five . so you might say well maybe we just subtract corresponding entries . and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five . and if you work out the math you 're going to get the exact same result as just subtracting the corresponding terms . so this is going to be -what is this going to be ? zero minus negative one is positive one , one minus three is negative two , three minus zero is three , two minus five is negative three . and you 'll see that you get the exact same thing here . when you multiply negative one times negative one you get positive one , positive one plus zero is one . negative one times three plus one is negative two . fair enough . there might be a question that is lingering in your brain right now . `` okay sal , i understand when i 'm adding or subtracting matrices with the same dimensions i just add or subtract the corresponding terms . but what happens when i have matrices with different dimensions ? '' so , for example , what about the scenario where i want to add the matrix one , zero , three , five , zero , one to the matrix -so this a three-by-two matrix- and i wan na add it to , let 's say , a two-by-two matrix . five , seven , negative one , zero . what would we define this as ? well it turns out that the mathematical mainstream does not define this . this is undefined . this is undefined . so we do not define matrix addition , or matrix subtraction , when the matrices have different dimensions . there did n't seem to be any reasonable way to do this , that would actually be useful and logically consistent in some nice way .
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and that indeed is how you can define matrix subtraction . in fact you do n't even have to define matrix subtraction , you can let this fall out of what we did with scalar multiplication and matrix addition . we can view as the exact same thing -this as the exact same thing- as taking zero , one , three , two and to that we add negative one , negative one times negative one , three , zero , five .
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the identity matrix can serve as a scalar , so why ca n't composed matrices be converted to bigger sizes ?
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