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Another method of finding the inverse is by augmenting with the identity. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
We can augment a3Ã33Ã3matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Write the system of equations asAX=B,AX=B,and multiply both sides by the inverse ofA:Aâ1AX=Aâ1B.A:Aâ1AX=Aâ1B.SeeExample 7andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
We can also use a calculator to solve a system of equations with matrix inverses. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The determinant for[abcd][abcd]isadâbc.adâbc.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Cramerâs Rule replaces a variable column with the constant column. Solutions arex=DxD,y=DyD.x=DxD,y=DyD.SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
To find the determinant of a 3Ã3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
To solve a system of three equations in three variables using Cramerâs Rule, replace a variable column with the constant column for each desired solution:x=DxD,y=DyD,z=DzD.x=DxD,y=DyD,z=DzD.SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Cramerâs Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Certain properties of determinants are useful for solving problems. For example:If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.When two rows are interchanged, the determinant changes sign.If either two rows or two columns are identical, the determinant eq... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
When two rows are interchanged, the determinant changes sign. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
If either two rows or two columns are identical, the determinant equals zero. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The determinant of an inverse matrixAâ1Aâ1is the reciprocal of the determinant of the matrixA.A. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. SeeExample 7andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
I 2 = [ 1 0 0 1 ] I 2 = [ 1 0 0 1 ] | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-equations |
I 3 = [ 1 0 0 0 1 0 0 0 1 ] I 3 = [ 1 0 0 0 1 0 0 0 1 ] | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-equations |
A â1 = 1 a d â b c [ d â b â c a ] , where a d â b c â 0 A â1 = 1 a d â b c [ d â b â c a ] , where a d â b c â 0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-equations |
addition method : an algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
augmented matrix : a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
break-even point : the point at which a cost function intersects a revenue function; where profit is zero | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
coefficient matrix : a matrix that contains only the coefficients from a system of equations | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
column : a set of numbers aligned vertically in a matrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
consistent system : a system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
cost function : the function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
Cramerâs Rule : a method for solving systems of equations that have the same number of equations as variables using determinants | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
dependent system : a system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
determinant : a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
entry : an element, coefficient, or constant in a matrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
feasible region : the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
Gaussian elimination : using elementary row operations to obtain a matrix in row-echelon form | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
identity matrix : a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
inconsistent system : a system of linear equations with no common solution because they represent parallel lines, which have no point or line in common | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
independent system : a system of linear equations with exactly one solution pair(x,y)(x,y) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
main diagonal : entries from the upper left corner diagonally to the lower right corner of a square matrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
matrix : a rectangular array of numbers | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
multiplicative inverse of a matrix : a matrix that, when multiplied by the original, equals the identity matrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
nonlinear inequality : an inequality containing a nonlinear expression | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
partial fraction decomposition : the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
partial fractions : the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
profit function : the profit function is written asP(x)=R(x)âC(x),P(x)=R(x)âC(x),revenue minus cost | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
revenue function : the function that is used to calculate revenue, simply written asR=xp,R=xp,wherex=x=quantity andp=p=price | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
row : a set of numbers aligned horizontally in a matrix | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
row operations : adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
row-echelon form : after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
row-equivalent : two matricesAAandBBare row-equivalent if one can be obtained from the other by performing basic row operations | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
scalar multiple : an entry of a matrix that has been multiplied by a scalar | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
solution set : the set of all ordered pairs or triples that satisfy all equations in a system of equations | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
substitution method : an algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
system of linear equations : a set of two or more equations in two or more variables that must be considered simultaneously. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
system of nonlinear equations : a system of equations containing at least one equation that is of degree larger than one | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
system of nonlinear inequalities : a system of two or more inequalities in two or more variables containing at least one inequality that is not linear | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-terms |
An ellipse is the set of all points(x,y)(x,y)in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci). | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. SeeExample 1andExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
A hyperbola is the set of all points(x,y)(x,y)in a plane such that the difference of the distances between(x,y)(x,y)and the foci is a positive constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The standard form of a hyperbola can be used to locate its vertices and foci. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
A parabola is the set of all points(x,y)(x,y)in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The standard form of a parabola with vertex(0,0)(0,0)and thex-axis as its axis of symmetry can be used to graph the parabola. Ifp>0,p>0,the parabola opens right. Ifp<0,p<0,the parabola opens left. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The standard form of a parabola with vertex(0,0)(0,0)and they-axis as its axis of symmetry can be used to graph the parabola. Ifp>0,p>0,the parabola opens up. Ifp<0,p<0,the parabola opens down. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
When given the focus and directrix of a parabola, we can write its equation in standard form. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The standard form of a parabola with vertex(h,k)(h,k)and axis of symmetry parallel to thex-axis can be used to graph the parabola. Ifp>0,p>0,the parabola opens right. Ifp<0,p<0,the parabola opens left. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The standard form of a parabola with vertex(h,k)(h,k)and axis of symmetry parallel to they-axis can be used to graph the parabola. Ifp>0,p>0,the parabola opens up. Ifp<0,p<0,the parabola opens down. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
A nondegenerate conic section has the general formAx2+Bxy+Cy2+Dx+Ey+F=0Ax2+Bxy+Cy2+Dx+Ey+F=0whereA,BA,BandCCare not all zero. The values ofA,B,A,B,andCCdetermine the type of conic. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Equations of conic sections with anxyxyterm have been rotated about the origin. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The general form can be transformed into an equation in thexâ²xâ²andyâ²yâ²coordinate system without thexâ²yâ²xâ²yâ²term. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focusP(r,θ)P(r,θ)at the pole, and a line, the directrix, which is perpendicular to the polar axis. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
A conic is the set of all pointse=PFPD,e=PFPD,where eccentricityeeis a positive real number. Each conic may be written in terms of its polar equation. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
The polar equations of conics can be graphed. SeeExample 2,Example 3, andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
Conics can be defined in terms of a focus, a directrix, and eccentricity. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
We can use the identitiesr=x2+y2,x=rcosθ,r=x2+y2,x=rcosθ,andy=rsinθy=rsinθto convert the equation for a conic from polar to rectangular form. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-concepts |
x 2 a 2 + y 2 b 2 = 1 , a > b x 2 a 2 + y 2 b 2 = 1 , a > b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
x 2 b 2 + y 2 a 2 = 1 , a > b x 2 b 2 + y 2 a 2 = 1 , a > b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( x â h ) 2 a 2 + ( y â k ) 2 b 2 = 1 , a > b ( x â h ) 2 a 2 + ( y â k ) 2 b 2 = 1 , a > b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( x â h ) 2 b 2 + ( y â k ) 2 a 2 = 1 , a > b ( x â h ) 2 b 2 + ( y â k ) 2 a 2 = 1 , a > b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
x 2 a 2 â y 2 b 2 = 1 x 2 a 2 â y 2 b 2 = 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
y 2 a 2 â x 2 b 2 = 1 y 2 a 2 â x 2 b 2 = 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( x â h ) 2 a 2 â ( y â k ) 2 b 2 = 1 ( x â h ) 2 a 2 â ( y â k ) 2 b 2 = 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( y â k ) 2 a 2 â ( x â h ) 2 b 2 = 1 ( y â k ) 2 a 2 â ( x â h ) 2 b 2 = 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
y 2 = 4 p x y 2 = 4 p x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
x 2 = 4 p y x 2 = 4 p y | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( y â k ) 2 = 4 p ( x â h ) ( y â k ) 2 = 4 p ( x â h ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
( x â h ) 2 = 4 p ( y â k ) ( x â h ) 2 = 4 p ( y â k ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
A x 2 + B x y + C y 2 + D x + E y + F = 0 A x 2 + B x y + C y 2 + D x + E y + F = 0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
x = x â² cos θ â y â² sin θ y = x â² sin θ + y â² cos θ x = x â² cos θ â y â² sin θ y = x â² sin θ + y â² cos θ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
θ , where cot ( 2 θ ) = A â C B θ , where cot ( 2 θ ) = A â C B | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-equations |
angle of rotation : an acute angle formed by a set of axes rotated from the Cartesian plane where, ifcot(2θ)>0,cot(2θ)>0,thenθθis between(0°,45°);(0°,45°);ifcot(2θ)<0,cot(2θ)<0,thenθθis between(45°,90°);(45°,90°);and ifcot(2θ)=0,cot(2θ)=0,thenθ=45°θ=45° | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
center of a hyperbola : the midpoint of both the transverse and conjugate axes of a hyperbola | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
center of an ellipse : the midpoint of both the major and minor axes | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
conic section : any shape resulting from the intersection of a right circular cone with a plane | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
conjugate axis : the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints | https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-key-terms |
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