diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1095.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1095.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2eab2968f24c58a033eb50c1673eb91c0b9b00d --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1095.txt @@ -0,0 +1,549 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-1)+(-1)\, \times \, 0+(-1)\, \times \, 1=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$2$} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-1)+(-1)\, (-1)+(-1)\, (-2)=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & \fbox{$6$} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-2)+(-1)\, \times \, 1+(-1)\, \times \, 3=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & \fbox{$2$} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+2\ 0+2\ 1=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + \fbox{$4$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+2 (-1)+2 (-2)=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & \fbox{$-4$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+2\ 1+2\ 3=12. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & \fbox{$12$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+1\ 0+(-2)\, \times \, 1=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + \fbox{$0$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + 0 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+1 (-1)+(-2)\, (-2)=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + 0 & \fbox{$5$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + 0 & 5 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+1\ 1+(-2)\, \times \, 3=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -3 & -1 & -1 \\ + -2 & 2 & 2 \\ + -2 & 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & -1 & -2 \\ + 0 & -1 & 1 \\ + 1 & -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 2 & 6 & 2 \\ + 4 & -4 & 12 \\ + 0 & 5 & \fbox{$-1$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1159.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1159.txt new file mode 100644 index 0000000000000000000000000000000000000000..91bafc07edcb61e274891ad7608aaa239e51b39e --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1159.txt @@ -0,0 +1,528 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{31\ 15}{16\ 8}+\frac{21 (-43)}{8\ 16}=-\frac{219}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-\frac{219}{64}$} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{31\ 45}{16\ 16}+\frac{21 (-25)}{8\ 16}=\frac{345}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \fbox{$\frac{345}{256}$} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{31 (-31)}{16\ 16}+\frac{21 (-3)}{8\ 2}=-\frac{1969}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & \fbox{$-\frac{1969}{256}$} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{23\ 15}{8\ 8}+\left(-\frac{39}{16}\right)\, \left(-\frac{43}{16}\right)=\frac{3057}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \fbox{$\frac{3057}{256}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{23\ 45}{8\ 16}+\left(-\frac{39}{16}\right)\, \left(-\frac{25}{16}\right)=\frac{3045}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \fbox{$\frac{3045}{256}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{23 (-31)}{8\ 16}+\left(-\frac{39}{16}\right)\, \left(-\frac{3}{2}\right)=-\frac{245}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & \fbox{$-\frac{245}{128}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{9\ 15}{16\ 8}+\frac{3 (-43)}{4\ 16}=-\frac{123}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + \fbox{$-\frac{123}{128}$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + -\frac{123}{128} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{9\ 45}{16\ 16}+\frac{3 (-25)}{4\ 16}=\frac{105}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + -\frac{123}{128} & \fbox{$\frac{105}{256}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + -\frac{123}{128} & \frac{105}{256} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{9 (-31)}{16\ 16}+\frac{3 (-3)}{4\ 2}=-\frac{567}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{31}{16} & \frac{21}{8} \\ + \frac{23}{8} & -\frac{39}{16} \\ + \frac{9}{16} & \frac{3}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{15}{8} & \frac{45}{16} & -\frac{31}{16} \\ + -\frac{43}{16} & -\frac{25}{16} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{219}{64} & \frac{345}{256} & -\frac{1969}{256} \\ + \frac{3057}{256} & \frac{3045}{256} & -\frac{245}{128} \\ + -\frac{123}{128} & \frac{105}{256} & \fbox{$-\frac{567}{256}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1226.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1226.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d6a977d43dddaa5eb76c10f626fb38264f5b1e3 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1226.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+2 (-2)=-6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-6$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+2\ 3=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & \fbox{$4$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 4 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+1 (-2)=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 4 \\ + \fbox{$0$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 4 \\ + 0 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+1\ 3=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + 2 & 2 \\ + -2 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -1 \\ + -2 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 4 \\ + 0 & \fbox{$5$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1239.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1239.txt new file mode 100644 index 0000000000000000000000000000000000000000..aaed3505a338fb646632fd186a27d7727712c171 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1239.txt @@ -0,0 +1,231 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 2+3\ 1+2 (-2)=3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$3$} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 3 \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 2+0\ 1+1 (-2)=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 3 \\ + \fbox{$2$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 3 \\ + 2 \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 2+2\ 1+0 (-2)=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 2 & 3 & 2 \\ + 2 & 0 & 1 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 1 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 3 \\ + 2 \\ + \fbox{$-2$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1285.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1285.txt new file mode 100644 index 0000000000000000000000000000000000000000..04a9ee1d89337691e18a376c29cf1f290903b987 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1285.txt @@ -0,0 +1,159 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 1+(-3)\, \times \, 3=-10. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$-10$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + -10 \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 1+2\ 3=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -1 & -3 \\ + -1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + -10 \\ + \fbox{$5$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1485.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1485.txt new file mode 100644 index 0000000000000000000000000000000000000000..9431d9c4639a68746ef6ddf319762c7c8d412bdf --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1485.txt @@ -0,0 +1,390 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{15}{8}\right)\, \times \, \frac{39}{16}+\frac{19\ 19}{8\ 16}+\left(-\frac{29}{16}\right)\, \left(-\frac{21}{8}\right)=\frac{385}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$\frac{385}{128}$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{15}{8}\right)\, \left(-\frac{7}{8}\right)+\frac{19\ 13}{8\ 8}+\left(-\frac{29}{16}\right)\, \left(-\frac{3}{2}\right)=\frac{263}{32}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \fbox{$\frac{263}{32}$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3\ 39}{16}+\frac{17\ 19}{16\ 16}+\left(-\frac{13}{8}\right)\, \left(-\frac{21}{8}\right)=\frac{3287}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \fbox{$\frac{3287}{256}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-7)}{8}+\frac{17\ 13}{16\ 8}+\left(-\frac{13}{8}\right)\, \left(-\frac{3}{2}\right)=\frac{197}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \fbox{$\frac{197}{128}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \frac{197}{128} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, \frac{39}{16}+\left(-\frac{13}{8}\right)\, \times \, \frac{19}{16}+\left(-\frac{5}{4}\right)\, \left(-\frac{21}{8}\right)=-\frac{451}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \frac{197}{128} \\ + \fbox{$-\frac{451}{128}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \frac{197}{128} \\ + -\frac{451}{128} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \left(-\frac{7}{8}\right)+\left(-\frac{13}{8}\right)\, \times \, \frac{13}{8}+\left(-\frac{5}{4}\right)\, \left(-\frac{3}{2}\right)=\frac{63}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{15}{8} & \frac{19}{8} & -\frac{29}{16} \\ + 3 & \frac{17}{16} & -\frac{13}{8} \\ + -2 & -\frac{13}{8} & -\frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{39}{16} & -\frac{7}{8} \\ + \frac{19}{16} & \frac{13}{8} \\ + -\frac{21}{8} & -\frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{385}{128} & \frac{263}{32} \\ + \frac{3287}{256} & \frac{197}{128} \\ + -\frac{451}{128} & \fbox{$\frac{63}{64}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1567.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1567.txt new file mode 100644 index 0000000000000000000000000000000000000000..db237b9509e0bdbf288723b43ed7300c42268896 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1567.txt @@ -0,0 +1,362 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{6}\right)\, \times \, \frac{5}{2}+\frac{8\ 3}{3}+\left(-\frac{7}{3}\right)\, \times \, \frac{11}{6}=\frac{29}{36}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$\frac{29}{36}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{6}\right)\, \left(-\frac{7}{3}\right)+\frac{8 (-3)}{3\ 2}+\left(-\frac{7}{3}\right)\, \left(-\frac{4}{3}\right)=\frac{11}{6}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \fbox{$\frac{11}{6}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{6}\right)\, (-3)+\frac{8 (-2)}{3}+\left(-\frac{7}{3}\right)\, \times \, \frac{8}{3}=-\frac{145}{18}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & \fbox{$-\frac{145}{18}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 5}{3\ 2}+\left(-\frac{3}{2}\right)\, \times \, 3+\frac{11}{6}=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + \fbox{$-1$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + -1 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2 (-7)}{3\ 3}+\left(-\frac{3}{2}\right)\, \left(-\frac{3}{2}\right)-\frac{4}{3}=-\frac{23}{36}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + -1 & \fbox{$-\frac{23}{36}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + -1 & -\frac{23}{36} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2 (-3)}{3}+\left(-\frac{3}{2}\right)\, (-2)+\frac{8}{3}=\frac{11}{3}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{7}{6} & \frac{8}{3} & -\frac{7}{3} \\ + \frac{2}{3} & -\frac{3}{2} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + \frac{5}{2} & -\frac{7}{3} & -3 \\ + 3 & -\frac{3}{2} & -2 \\ + \frac{11}{6} & -\frac{4}{3} & \frac{8}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{29}{36} & \frac{11}{6} & -\frac{145}{18} \\ + -1 & -\frac{23}{36} & \fbox{$\frac{11}{3}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1633.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1633.txt new file mode 100644 index 0000000000000000000000000000000000000000..9de0386228a13092bbb9564ac05cd486a6548fa2 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1633.txt @@ -0,0 +1,549 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+(-1)\, \times \, 0+2\ 3=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$4$} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 3+(-1)\, \times \, 2+2\ 1=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & \fbox{$6$} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+(-1)\, \times \, 2+2 (-2)=-10. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & \fbox{$-10$} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-1)+(-1)\, \times \, 0+0\ 3=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + \fbox{$-1$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 3+(-1)\, \times \, 2+0\ 1=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & \fbox{$1$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-2)+(-1)\, \times \, 2+0 (-2)=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & \fbox{$-4$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+2\ 0+0\ 3=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + \fbox{$-2$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + -2 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 3+2\ 2+0\ 1=10. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + -2 & \fbox{$10$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + -2 & 10 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+2\ 2+0 (-2)=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 2 & -1 & 2 \\ + 1 & -1 & 0 \\ + 2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 3 & -2 \\ + 0 & 2 & 2 \\ + 3 & 1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 4 & 6 & -10 \\ + -1 & 1 & -4 \\ + -2 & 10 & \fbox{$0$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1703.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1703.txt new file mode 100644 index 0000000000000000000000000000000000000000..3feb372cfcf25ffcdc2e11a81a5116395f4d1df6 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1703.txt @@ -0,0 +1,231 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{5}{2}\right)\, \left(-\frac{3}{4}\right)+\frac{1}{2\ 2}+\left(-\frac{5}{2}\right)\, \times \, \frac{1}{2}=\frac{7}{8}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$\frac{7}{8}$} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{7}{8} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }-\frac{3}{4}+\frac{3}{2\ 2}+\frac{1}{2}=\frac{1}{2}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{7}{8} \\ + \fbox{$\frac{1}{2}$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{7}{8} \\ + \frac{1}{2} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{5}{4}\right)\, \left(-\frac{3}{4}\right)+\frac{1}{2\ 2}+\frac{9}{4\ 2}=\frac{37}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{5}{2} & \frac{1}{2} & -\frac{5}{2} \\ + 1 & \frac{3}{2} & 1 \\ + -\frac{5}{4} & \frac{1}{2} & \frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{c} + -\frac{3}{4} \\ + \frac{1}{2} \\ + \frac{1}{2} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{7}{8} \\ + \frac{1}{2} \\ + \fbox{$\frac{37}{16}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/1934.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/1934.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc189d2bddac536088f3c99bf786be2f61f6d097 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/1934.txt @@ -0,0 +1,264 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0 (-3)+1\ 1+2\ 2=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$5$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0 (-1)+1 (-1)+2 (-2)=-5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & \fbox{$-5$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & -5 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-3)+(-2)\, \times \, 1+(-2)\, \times \, 2=3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & -5 \\ + \fbox{$3$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & -5 \\ + 3 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-1)+(-2)\, (-1)+(-2)\, (-2)=9. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 0 & 1 & 2 \\ + -3 & -2 & -2 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -3 & -1 \\ + 1 & -1 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 5 & -5 \\ + 3 & \fbox{$9$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2027.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2027.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e97744bbeb1f7b72883606a30b0c6ddfcae8c8b --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2027.txt @@ -0,0 +1,222 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{9\ 3}{16\ 8}+\left(-\frac{3}{8}\right)\, \left(-\frac{13}{8}\right)=\frac{105}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$\frac{105}{128}$} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{105}{128} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3}{16\ 8}+\frac{35 (-13)}{16\ 8}=-\frac{113}{32}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{105}{128} \\ + \fbox{$-\frac{113}{32}$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{105}{128} \\ + -\frac{113}{32} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5\ 3}{8\ 8}+\frac{7 (-13)}{8\ 8}=-\frac{19}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{9}{16} & -\frac{3}{8} \\ + \frac{1}{16} & \frac{35}{16} \\ + \frac{5}{8} & \frac{7}{8} \\ +\end{array} +\right).\left( +\begin{array}{c} + \frac{3}{8} \\ + -\frac{13}{8} \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{105}{128} \\ + -\frac{113}{32} \\ + \fbox{$-\frac{19}{16}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2119.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2119.txt new file mode 100644 index 0000000000000000000000000000000000000000..8762280124790f963d0ad0e55923513c9cb9db4f --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2119.txt @@ -0,0 +1,159 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 3+1\ 1=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$-2$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)=\left( +\begin{array}{c} + -2 \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 3+2\ 1=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -1 & 1 \\ + 1 & 2 \\ +\end{array} +\right).\left( +\begin{array}{c} + 3 \\ + 1 \\ +\end{array} +\right)=\left( +\begin{array}{c} + -2 \\ + \fbox{$5$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2230.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2230.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba4d7493a9256a91d5c240a36ed1a57a699ca15d --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2230.txt @@ -0,0 +1,390 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{8}\right)\, \left(-\frac{27}{16}\right)+\frac{3\ 3}{2\ 2}+\frac{29\ 27}{16\ 16}=\frac{1845}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$\frac{1845}{256}$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{8}\right)\, \times \, \frac{19}{16}+\frac{3\ 43}{2\ 16}+\frac{29\ 3}{16\ 2}=\frac{693}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \fbox{$\frac{693}{128}$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{33 (-27)}{16\ 16}+\frac{3}{8\ 2}+\frac{23\ 27}{16\ 16}=-\frac{111}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + \fbox{$-\frac{111}{128}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{33\ 19}{16\ 16}+\frac{43}{8\ 16}+\frac{23\ 3}{16\ 2}=\frac{1265}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \fbox{$\frac{1265}{256}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \frac{1265}{256} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{5}{2}\right)\, \left(-\frac{27}{16}\right)+\frac{19\ 3}{8\ 2}+\frac{5\ 27}{4\ 16}=\frac{633}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \frac{1265}{256} \\ + \fbox{$\frac{633}{64}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \frac{1265}{256} \\ + \frac{633}{64} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{5}{2}\right)\, \times \, \frac{19}{16}+\frac{19\ 43}{8\ 16}+\frac{5\ 3}{4\ 2}=\frac{677}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{9}{8} & \frac{3}{2} & \frac{29}{16} \\ + \frac{33}{16} & \frac{1}{8} & \frac{23}{16} \\ + -\frac{5}{2} & \frac{19}{8} & \frac{5}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{27}{16} & \frac{19}{16} \\ + \frac{3}{2} & \frac{43}{16} \\ + \frac{27}{16} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{1845}{256} & \frac{693}{128} \\ + -\frac{111}{128} & \frac{1265}{256} \\ + \frac{633}{64} & \fbox{$\frac{677}{128}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/231.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/231.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7f957d0e60b164947c84e76acb29955b985e263 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/231.txt @@ -0,0 +1,390 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+(-2)\, \times \, 2+(-3)\, \times \, 2=-6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-6$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 0+(-2)\, \times \, 0+(-3)\, \times \, 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & \fbox{$0$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, (-2)+1\ 2+0\ 2=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + \fbox{$4$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 0+1\ 0+0\ 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & \fbox{$0$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & 0 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+2\ 2+(-1)\, \times \, 2=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & 0 \\ + \fbox{$6$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & 0 \\ + 6 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 0+2\ 0+(-1)\, \times \, 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -2 & -2 & -3 \\ + -1 & 1 & 0 \\ + -2 & 2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -2 & 0 \\ + 2 & 0 \\ + 2 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -6 & 0 \\ + 4 & 0 \\ + 6 & \fbox{$0$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2648.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2648.txt new file mode 100644 index 0000000000000000000000000000000000000000..694b2a4d084f7e199951f6936b3be19ccfd6c2fa --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2648.txt @@ -0,0 +1,549 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 0+2\ 0+(-1)\, \times \, 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$0$} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 2+2 (-1)+(-1)\, (-1)=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & \fbox{$-1$} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 1+2 (-1)+(-1)\, \times \, 0=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & \fbox{$-2$} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 0+0\ 0+(-1)\, \times \, 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + \fbox{$0$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 2+0 (-1)+(-1)\, (-1)=-5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & \fbox{$-5$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 1+0 (-1)+(-1)\, \times \, 0=-3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & \fbox{$-3$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 0+(-1)\, \times \, 0+1\ 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + \fbox{$0$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + 0 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 2+(-1)\, (-1)+1 (-1)=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + 0 & \fbox{$6$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + 0 & 6 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 1+(-1)\, (-1)+1\ 0=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 0 & 2 & -1 \\ + -3 & 0 & -1 \\ + 3 & -1 & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 0 & 2 & 1 \\ + 0 & -1 & -1 \\ + 0 & -1 & 0 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & -5 & -3 \\ + 0 & 6 & \fbox{$4$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2680.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2680.txt new file mode 100644 index 0000000000000000000000000000000000000000..7521908866a6830176630fe15a7b5938369a6fa0 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2680.txt @@ -0,0 +1,231 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 2+(-2)\, \times \, 0+(-1)\, (-2)=8. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$8$} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 8 \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 2+3\ 0+(-3)\, (-2)=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 8 \\ + \fbox{$2$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 8 \\ + 2 \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 2+1\ 0+(-1)\, (-2)=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 3 & -2 & -1 \\ + -2 & 3 & -3 \\ + -2 & 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 2 \\ + 0 \\ + -2 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 8 \\ + 2 \\ + \fbox{$-2$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2701.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2701.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc7ef9448d57118cecbaa87190b967a0a608ed72 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2701.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 1+(-2)\, \times \, 2=-3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-3$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 0+(-2)\, (-3)=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & \fbox{$6$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & 6 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 1+3\ 2=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & 6 \\ + \fbox{$4$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & 6 \\ + 4 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 0+3 (-3)=-9. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + 1 & -2 \\ + -2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & 0 \\ + 2 & -3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -3 & 6 \\ + 4 & \fbox{$-9$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/274.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/274.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f5119415c7f17c229c8a17802675a5fe7ff477b --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/274.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-11)}{2\ 4}+\frac{9}{4\ 4}=-\frac{57}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-\frac{57}{16}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{4} \left(-\frac{3}{2}\right)+\frac{1}{4}=-\frac{1}{8}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & \fbox{$-\frac{1}{8}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & -\frac{1}{8} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7 (-11)}{4\ 4}+\left(-\frac{9}{4}\right)\, \left(-\frac{9}{4}\right)=\frac{1}{4}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & -\frac{1}{8} \\ + \fbox{$\frac{1}{4}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & -\frac{1}{8} \\ + \frac{1}{4} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{4} \left(-\frac{7}{4}\right)+\left(-\frac{9}{4}\right)\, (-1)=\frac{29}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{3}{2} & -\frac{1}{4} \\ + \frac{7}{4} & -\frac{9}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + -\frac{11}{4} & -\frac{1}{4} \\ + -\frac{9}{4} & -1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{57}{16} & -\frac{1}{8} \\ + \frac{1}{4} & \fbox{$\frac{29}{16}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2741.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2741.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f826b98c6087fa598193ce4868b01f35abc0ab2 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2741.txt @@ -0,0 +1,390 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 1+(-1)\, \times \, 0+1 (-2)=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$0$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+(-1)\, \times \, 0+1\ 1=-3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & \fbox{$-3$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 1+3\ 0+(-1)\, (-2)=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + \fbox{$1$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, (-2)+3\ 0+(-1)\, \times \, 1=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & \fbox{$1$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & 1 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 1+1\ 0+3 (-2)=-7. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & 1 \\ + \fbox{$-7$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & 1 \\ + -7 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, (-2)+1\ 0+3\ 1=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 2 & -1 & 1 \\ + -1 & 3 & -1 \\ + -1 & 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 1 & -2 \\ + 0 & 0 \\ + -2 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 0 & -3 \\ + 1 & 1 \\ + -7 & \fbox{$5$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/2863.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/2863.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9a6b181bbcd8e48f21f1ee54bb64258021623f8 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/2863.txt @@ -0,0 +1,362 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+(-3)\, \times \, 3+0 (-1)=-13. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-13$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 1+(-3)\, (-2)+0 (-1)=8. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & \fbox{$8$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+(-3)\, \times \, 1+0 (-2)=-7. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & \fbox{$-7$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+2\ 3+0 (-1)=10. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + \fbox{$10$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + 10 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 1+2 (-2)+0 (-1)=-6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + 10 & \fbox{$-6$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + 10 & -6 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+2\ 1+0 (-2)=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 2 & -3 & 0 \\ + -2 & 2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -2 & 1 & -2 \\ + 3 & -2 & 1 \\ + -1 & -1 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -13 & 8 & -7 \\ + 10 & -6 & \fbox{$6$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3051.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3051.txt new file mode 100644 index 0000000000000000000000000000000000000000..396913c081a45be9920a6923ae22cb00cc367d07 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3051.txt @@ -0,0 +1,159 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-1)+2\ 3=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$5$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 5 \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-1)+(-2)\, \times \, 3=-7. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + 1 & 2 \\ + 1 & -2 \\ +\end{array} +\right).\left( +\begin{array}{c} + -1 \\ + 3 \\ +\end{array} +\right)=\left( +\begin{array}{c} + 5 \\ + \fbox{$-7$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3179.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3179.txt new file mode 100644 index 0000000000000000000000000000000000000000..fecf3afef3aa890babbd02afec3289df3557aef2 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3179.txt @@ -0,0 +1,347 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{43 (-3)}{16\ 4}+\frac{3 (-7)}{2\ 4}=-\frac{297}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-\frac{297}{64}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{43 (-41)}{16\ 16}+\frac{3 (-23)}{2\ 8}=-\frac{2867}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & \fbox{$-\frac{2867}{256}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{43\ 7}{16\ 8}+\frac{3\ 23}{2\ 16}=\frac{577}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \fbox{$\frac{577}{128}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{45}{16}\right)\, \left(-\frac{3}{4}\right)+\frac{11 (-7)}{4\ 4}=-\frac{173}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + \fbox{$-\frac{173}{64}$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + -\frac{173}{64} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{45}{16}\right)\, \left(-\frac{41}{16}\right)+\frac{11 (-23)}{4\ 8}=-\frac{179}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + -\frac{173}{64} & \fbox{$-\frac{179}{256}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + -\frac{173}{64} & -\frac{179}{256} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{45}{16}\right)\, \times \, \frac{7}{8}+\frac{11\ 23}{4\ 16}=\frac{191}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{43}{16} & \frac{3}{2} \\ + -\frac{45}{16} & \frac{11}{4} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{3}{4} & -\frac{41}{16} & \frac{7}{8} \\ + -\frac{7}{4} & -\frac{23}{8} & \frac{23}{16} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{297}{64} & -\frac{2867}{256} & \frac{577}{128} \\ + -\frac{173}{64} & -\frac{179}{256} & \fbox{$\frac{191}{128}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3347.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3347.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae13be87dcc6435bb262cbca22ca098e42e7ad88 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3347.txt @@ -0,0 +1,264 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{6}\right)\, \times \, 2+\left(-\frac{8}{3}\right)\, \left(-\frac{7}{6}\right)+\frac{4}{2\ 3}=\frac{13}{9}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$\frac{13}{9}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{6}\right)\, \left(-\frac{5}{6}\right)+\left(-\frac{8}{3}\right)\, (-2)-\frac{5}{2\ 3}=\frac{197}{36}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \fbox{$\frac{197}{36}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \frac{197}{36} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{6} (-2)-\frac{7}{3\ 6}+\frac{4}{6\ 3}=-\frac{1}{2}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \frac{197}{36} \\ + \fbox{$-\frac{1}{2}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \frac{197}{36} \\ + -\frac{1}{2} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5}{6\ 6}-\frac{2}{3}-\frac{5}{6\ 3}=-\frac{29}{36}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{7}{6} & -\frac{8}{3} & \frac{1}{2} \\ + -\frac{1}{6} & \frac{1}{3} & \frac{1}{6} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & -\frac{5}{6} \\ + -\frac{7}{6} & -2 \\ + \frac{4}{3} & -\frac{5}{3} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{13}{9} & \frac{197}{36} \\ + -\frac{1}{2} & \fbox{$-\frac{29}{36}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3354.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3354.txt new file mode 100644 index 0000000000000000000000000000000000000000..72e20ba4d40fb06eae481876280cb8e98bbdd555 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3354.txt @@ -0,0 +1,347 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+1\ 1=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-1$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 1+1\ 1=3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & \fbox{$3$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+1\ 1=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & \fbox{$-1$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+(-1)\, \times \, 1=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + \fbox{$1$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + 1 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 1+(-1)\, \times \, 1=-3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + 1 & \fbox{$-3$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + 1 & -3 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+(-1)\, \times \, 1=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + 2 & 1 \\ + -2 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 1 & -1 \\ + 1 & 1 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -1 & 3 & -1 \\ + 1 & -3 & \fbox{$1$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3426.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3426.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e160e9094b0b55694c144c054613fec9634f75c --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3426.txt @@ -0,0 +1,347 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4 (-2)}{3\ 3}+\frac{2 (-2)}{3}=-\frac{20}{9}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-\frac{20}{9}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4 (-2)}{3}+\frac{2\ 3}{3}=-\frac{2}{3}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & \fbox{$-\frac{2}{3}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4\ 7}{3\ 3}+\frac{2 (-2)}{3\ 3}=\frac{8}{3}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \fbox{$\frac{8}{3}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{3}\right)\, \left(-\frac{2}{3}\right)+1 (-2)=-\frac{4}{9}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + \fbox{$-\frac{4}{9}$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + -\frac{4}{9} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{3}\right)\, (-2)+1\ 3=\frac{23}{3}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + -\frac{4}{9} & \fbox{$\frac{23}{3}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + -\frac{4}{9} & \frac{23}{3} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{3}\right)\, \times \, \frac{7}{3}-\frac{2}{3}=-\frac{55}{9}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{4}{3} & \frac{2}{3} \\ + -\frac{7}{3} & 1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{2}{3} & -2 & \frac{7}{3} \\ + -2 & 3 & -\frac{2}{3} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -\frac{20}{9} & -\frac{2}{3} & \frac{8}{3} \\ + -\frac{4}{9} & \frac{23}{3} & \fbox{$-\frac{55}{9}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/359.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/359.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5cd409a85cddee48871c68f88eccc8b8def3bed --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/359.txt @@ -0,0 +1,231 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 1. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 1: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \_ \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{2} (-0)+\frac{3}{2\ 2}+0\ 0=\frac{3}{4}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \fbox{$\frac{3}{4}$} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{3}{4} \\ + \_ \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 0+\frac{1}{2} (-3)+(-2)\, \times \, 0=-\frac{3}{2}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{3}{4} \\ + \fbox{$-\frac{3}{2}$} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{3}{4} \\ + -\frac{3}{2} \\ + \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 0+\frac{5}{2\ 2}+(-1)\, \times \, 0=\frac{5}{4}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{1}{2} & -\frac{3}{2} & 0 \\ + 3 & 3 & -2 \\ + 0 & -\frac{5}{2} & -1 \\ +\end{array} +\right).\left( +\begin{array}{c} + 0 \\ + -\frac{1}{2} \\ + 0 \\ +\end{array} +\right)=\left( +\begin{array}{c} + \frac{3}{4} \\ + -\frac{3}{2} \\ + \fbox{$\frac{5}{4}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3665.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3665.txt new file mode 100644 index 0000000000000000000000000000000000000000..626bff1f248dc522ee2a59bd0afc6ba7eaf58149 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3665.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13\ 2}{5\ 5}+\frac{12 (-4)}{5\ 5}=-\frac{22}{25}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-\frac{22}{25}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13\ 2}{5}+\frac{12\ 12}{5\ 5}=\frac{274}{25}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \fbox{$\frac{274}{25}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \frac{274}{25} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7\ 2}{5\ 5}+\left(-\frac{11}{5}\right)\, \left(-\frac{4}{5}\right)=\frac{58}{25}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \frac{274}{25} \\ + \fbox{$\frac{58}{25}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \frac{274}{25} \\ + \frac{58}{25} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7\ 2}{5}+\left(-\frac{11}{5}\right)\, \times \, \frac{12}{5}=-\frac{62}{25}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{13}{5} & \frac{12}{5} \\ + \frac{7}{5} & -\frac{11}{5} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{2}{5} & 2 \\ + -\frac{4}{5} & \frac{12}{5} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{22}{25} & \frac{274}{25} \\ + \frac{58}{25} & \fbox{$-\frac{62}{25}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3849.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3849.txt new file mode 100644 index 0000000000000000000000000000000000000000..35d224872396a573a34119befd668cc0cdc5dcc7 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3849.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 2+2\ 0=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-2$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, \times \, 2+2 (-2)=-6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & \fbox{$-6$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & -6 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 2+3\ 0=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & -6 \\ + \fbox{$2$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & -6 \\ + 2 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 2+3 (-2)=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -1 & 2 \\ + 1 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + 0 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & -6 \\ + 2 & \fbox{$-4$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/3856.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/3856.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fd4ed2d126660f914bc4752c0466be6ba0d65c1 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/3856.txt @@ -0,0 +1,375 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+0\ 2=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$2$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+0 (-2)=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & \fbox{$4$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-1)+1\ 2=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + \fbox{$1$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-2)+1 (-2)=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & \fbox{$-4$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & -4 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+0\ 2=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & -4 \\ + \fbox{$2$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & -4 \\ + 2 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+0 (-2)=4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -2 & 0 \\ + 1 & 1 \\ + -2 & 0 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 2 & 4 \\ + 1 & -4 \\ + 2 & \fbox{$4$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/39.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/39.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f39ab06fe945bef3ef890d24886ad398ced95dc --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/39.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+(-2)\, \times \, 3=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-4$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+(-2)\, \times \, 3=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & \fbox{$-2$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & -2 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-1)+3\ 3=7. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & -2 \\ + \fbox{$7$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & -2 \\ + 7 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+3\ 3=5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -2 & -2 \\ + 2 & 3 \\ +\end{array} +\right).\left( +\begin{array}{cc} + -1 & -2 \\ + 3 & 3 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -4 & -2 \\ + 7 & \fbox{$5$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4014.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4014.txt new file mode 100644 index 0000000000000000000000000000000000000000..34b1c8c24f2cb9864e9eabc1a8b5e334e2214d60 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4014.txt @@ -0,0 +1,528 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{7}\right)\, (-1)+\left(-\frac{4}{7}\right)\, \times \, \frac{8}{7}=\frac{10}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$\frac{10}{49}$} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \_ & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{7}\right)\, \times \, 0+\left(-\frac{4}{7}\right)\, \left(-\frac{8}{7}\right)=\frac{32}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \fbox{$\frac{32}{49}$} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & \_ \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{7}\right)\, \left(-\frac{12}{7}\right)+\left(-\frac{4}{7}\right)\, \times \, \frac{20}{7}=-\frac{8}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & \fbox{$-\frac{8}{49}$} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{18 (-1)}{7}+\left(-\frac{8}{7}\right)\, \times \, \frac{8}{7}=-\frac{190}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + \fbox{$-\frac{190}{49}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{18\ 0}{7}+\left(-\frac{8}{7}\right)\, \left(-\frac{8}{7}\right)=\frac{64}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \fbox{$\frac{64}{49}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{18 (-12)}{7\ 7}+\left(-\frac{8}{7}\right)\, \times \, \frac{20}{7}=-\frac{376}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & \fbox{$-\frac{376}{49}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{19}{7}\right)\, (-1)+\frac{20\ 8}{7\ 7}=\frac{293}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \fbox{$\frac{293}{49}$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \frac{293}{49} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{19}{7}\right)\, \times \, 0+\frac{20 (-8)}{7\ 7}=-\frac{160}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \frac{293}{49} & \fbox{$-\frac{160}{49}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \frac{293}{49} & -\frac{160}{49} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{19}{7}\right)\, \left(-\frac{12}{7}\right)+\frac{20\ 20}{7\ 7}=\frac{628}{49}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -\frac{6}{7} & -\frac{4}{7} \\ + \frac{18}{7} & -\frac{8}{7} \\ + -\frac{19}{7} & \frac{20}{7} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -1 & 0 & -\frac{12}{7} \\ + \frac{8}{7} & -\frac{8}{7} & \frac{20}{7} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{10}{49} & \frac{32}{49} & -\frac{8}{49} \\ + -\frac{190}{49} & \frac{64}{49} & -\frac{376}{49} \\ + \frac{293}{49} & -\frac{160}{49} & \fbox{$\frac{628}{49}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4161.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4161.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e5a2b510d82d4939cb509e909e3046041ab925c --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4161.txt @@ -0,0 +1,347 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 1+1 (-2)=-5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$-5$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 1+1 (-2)=-5. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & \fbox{$-5$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 0+1 (-1)=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & \fbox{$-1$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 1+(-1)\, (-2)=3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + \fbox{$3$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + 3 & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 1+(-1)\, (-2)=3. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + 3 & \fbox{$3$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + 3 & 3 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 0+(-1)\, (-1)=1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -3 & 1 \\ + 1 & -1 \\ +\end{array} +\right).\left( +\begin{array}{ccc} + 1 & 1 & 0 \\ + -2 & -2 & -1 \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + -5 & -5 & -1 \\ + 3 & 3 & \fbox{$1$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4599.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4599.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf02748479119267fa27ecbb57c166c13a0b7402 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4599.txt @@ -0,0 +1,362 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 3. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 3: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \_ & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7}{4\ 4}+\left(-\frac{5}{2}\right)\, \left(-\frac{7}{4}\right)+\frac{5}{4\ 4}=\frac{41}{8}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \fbox{$\frac{41}{8}$} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & \_ & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{4}\right)\, \times \, \frac{11}{4}+\left(-\frac{5}{2}\right)\, \times \, 1-\frac{5}{4\ 2}=-\frac{127}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & \fbox{$-\frac{127}{16}$} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \_ \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{4}\right)\, \left(-\frac{11}{4}\right)+\left(-\frac{5}{2}\right)\, \left(-\frac{3}{2}\right)-\frac{5}{4\ 2}=\frac{127}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \fbox{$\frac{127}{16}$} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \_ & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3}{4}+(-2)\, \left(-\frac{7}{4}\right)+\left(-\frac{5}{2}\right)\, \times \, \frac{5}{4}=\frac{9}{8}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \fbox{$\frac{9}{8}$} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \frac{9}{8} & \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, \frac{11}{4}+(-2)\, \times \, 1+\left(-\frac{5}{2}\right)\, \left(-\frac{5}{2}\right)=-4. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \frac{9}{8} & \fbox{$-4$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }3^{\text{rd}} \text{column}: \\ + \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \frac{9}{8} & -4 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \left(-\frac{11}{4}\right)+(-2)\, \left(-\frac{3}{2}\right)+\left(-\frac{5}{2}\right)\, \left(-\frac{5}{2}\right)=\frac{35}{2}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + -\frac{7}{4} & -\frac{5}{2} & \frac{1}{4} \\ + -3 & -2 & -\frac{5}{2} \\ +\end{array} +\right).\left( +\begin{array}{ccc} + -\frac{1}{4} & \frac{11}{4} & -\frac{11}{4} \\ + -\frac{7}{4} & 1 & -\frac{3}{2} \\ + \frac{5}{4} & -\frac{5}{2} & -\frac{5}{2} \\ +\end{array} +\right)=\left( +\begin{array}{ccc} + \frac{41}{8} & -\frac{127}{16} & \frac{127}{16} \\ + \frac{9}{8} & -4 & \fbox{$\frac{35}{2}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4613.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4613.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f326926aaac386ba11c529e2248405e8832b5bc --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4613.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{25\ 2}{16}+\left(-\frac{5}{16}\right)\, \left(-\frac{21}{16}\right)=\frac{905}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$\frac{905}{256}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{25\ 2}{16}+\left(-\frac{5}{16}\right)\, \times \, \frac{9}{8}=\frac{355}{128}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \fbox{$\frac{355}{128}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \frac{355}{128} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2}{2}+\left(-\frac{7}{4}\right)\, \left(-\frac{21}{16}\right)=\frac{211}{64}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \frac{355}{128} \\ + \fbox{$\frac{211}{64}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \frac{355}{128} \\ + \frac{211}{64} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2}{2}+\left(-\frac{7}{4}\right)\, \times \, \frac{9}{8}=-\frac{31}{32}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + \frac{25}{16} & -\frac{5}{16} \\ + \frac{1}{2} & -\frac{7}{4} \\ +\end{array} +\right).\left( +\begin{array}{cc} + 2 & 2 \\ + -\frac{21}{16} & \frac{9}{8} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \frac{905}{256} & \frac{355}{128} \\ + \frac{211}{64} & \fbox{$-\frac{31}{32}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4661.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4661.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e3fde1f2adb91fcdb18cc76d91e955bf5f277c6 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4661.txt @@ -0,0 +1,375 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }3\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }3\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{35}{16}\right)\, \times \, \frac{23}{16}+\left(-\frac{5}{16}\right)\, \times \, \frac{21}{8}=-\frac{1015}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-\frac{1015}{256}$} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & \_ \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{35}{16}\right)\, \times \, \frac{1}{2}+\left(-\frac{5}{16}\right)\, \times \, \frac{3}{2}=-\frac{25}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & \fbox{$-\frac{25}{16}$} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13\ 23}{16\ 16}+\frac{13\ 21}{16\ 8}=\frac{845}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \fbox{$\frac{845}{256}$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13}{16\ 2}+\frac{13\ 3}{16\ 2}=\frac{13}{8}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \fbox{$\frac{13}{8}$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \frac{13}{8} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{47}{16}\right)\, \times \, \frac{23}{16}+\left(-\frac{37}{16}\right)\, \times \, \frac{21}{8}=-\frac{2635}{256}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \frac{13}{8} \\ + \fbox{$-\frac{2635}{256}$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }3^{\text{rd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \frac{13}{8} \\ + -\frac{2635}{256} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{47}{16}\right)\, \times \, \frac{1}{2}+\left(-\frac{37}{16}\right)\, \times \, \frac{3}{2}=-\frac{79}{16}. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }3^{\text{rd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + -\frac{35}{16} & -\frac{5}{16} \\ + \frac{13}{16} & \frac{13}{16} \\ + -\frac{47}{16} & -\frac{37}{16} \\ +\end{array} +\right).\left( +\begin{array}{cc} + \frac{23}{16} & \frac{1}{2} \\ + \frac{21}{8} & \frac{3}{2} \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -\frac{1015}{256} & -\frac{25}{16} \\ + \frac{845}{256} & \frac{13}{8} \\ + -\frac{2635}{256} & \fbox{$-\frac{79}{16}$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/4856.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/4856.txt new file mode 100644 index 0000000000000000000000000000000000000000..95fbb188ed7c130ca6967197d0cbf64371bf5488 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/4856.txt @@ -0,0 +1,264 @@ +Problem: +Multiply +$\left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 3 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }3\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 0+(-3)\, (-3)+0\ 0=9. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$9$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 1+(-3)\, (-2)+0\ 1=6. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & \fbox{$6$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & 6 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 0+0 (-3)+(-1)\, \times \, 0=0. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & 6 \\ + \fbox{$0$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & 6 \\ + 0 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 1+0 (-2)+(-1)\, \times \, 1=-1. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{ccc} + 0 & -3 & 0 \\ + 0 & 0 & -1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 1 \\ + -3 & -2 \\ + 0 & 1 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + 9 & 6 \\ + 0 & \fbox{$-1$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/linear_algebra/multiply_w_steps/734.txt b/pretraining/mathematica/linear_algebra/multiply_w_steps/734.txt new file mode 100644 index 0000000000000000000000000000000000000000..9db70df4c423b4ab48d38dd970878fb0960ae420 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/multiply_w_steps/734.txt @@ -0,0 +1,253 @@ +Problem: +Multiply +$\left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right)$ and +$\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Multiply the following matrices}: \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right) \\ +\end{array} + \\ +\hline + +\begin{array}{l} + +\begin{array}{l} + \text{The }\text{dimensions }\text{of }\text{the }\text{first }\text{matrix }\text{are }2\times 2 \text{and }\text{the }\text{dimensions }\text{of }\text{the }\text{second }\text{matrix }\text{are }2\times 2. \\ + \text{This }\text{means }\text{the }\text{dimensions }\text{of }\text{the }\text{product }\text{are }2\times 2: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \_ & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 0+(-1)\, \times \, 2=-2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + \fbox{$-2$} & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }1^{\text{st}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & \_ \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 3+(-1)\, (-2)=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }1^{\text{st}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & \fbox{$2$} \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }1^{\text{st}} \text{column}: \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & 2 \\ + \_ & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 0+1\ 2=2. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }1^{\text{st}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & 2 \\ + \fbox{$2$} & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + \text{Highlight }\text{the }2^{\text{nd}} \text{row }\text{and }\text{the }2^{\text{nd}} \text{column}: \\ + \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & 2 \\ + 2 & \_ \\ +\end{array} +\right) \\ +\end{array} + \\ + +\begin{array}{l} + +\begin{array}{l} + \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 3+1 (-2)=7. \\ + \text{Place }\text{this }\text{number }\text{into }\text{the }2^{\text{nd}} \text{row }\text{and }2^{\text{nd}} \text{column }\text{of }\text{the }\text{product}: \\ +\end{array} + \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left( +\begin{array}{cc} + 0 & -1 \\ + 3 & 1 \\ +\end{array} +\right).\left( +\begin{array}{cc} + 0 & 3 \\ + 2 & -2 \\ +\end{array} +\right)=\left( +\begin{array}{cc} + -2 & 2 \\ + 2 & \fbox{$7$} \\ +\end{array} +\right) \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/10331.txt b/pretraining/mathematica/number_theory/multiplicative_order/10331.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8b786d332c008d1fa021fdd647b4ce800ba0c5d --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/10331.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $36^m \equiv 1 \pmod{335}$. +Answer: +$33$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/10834.txt b/pretraining/mathematica/number_theory/multiplicative_order/10834.txt new file mode 100644 index 0000000000000000000000000000000000000000..168e9330c06479c2cb40cabc74c837b9d0ab0e99 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/10834.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $938^m \equiv 1 \pmod{951}$. +Answer: +$316$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/13083.txt b/pretraining/mathematica/number_theory/multiplicative_order/13083.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd8d517a442f3038748025d9c8076e73a9114ae3 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/13083.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $219^m \equiv 1 \pmod{380}$. +Answer: +$18$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/13232.txt b/pretraining/mathematica/number_theory/multiplicative_order/13232.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0929f0c644f0875cd4ddf1940c5228716574b16 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/13232.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $242^m \equiv 1 \pmod{335}$. +Answer: +$132$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/13795.txt b/pretraining/mathematica/number_theory/multiplicative_order/13795.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5b21877360666ae7f79b434fc0ea51f7221df43 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/13795.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $34^m \equiv 1 \pmod{135}$. +Answer: +$18$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/16023.txt b/pretraining/mathematica/number_theory/multiplicative_order/16023.txt new file mode 100644 index 0000000000000000000000000000000000000000..548ce649482f4e681f3e118f4a677a29a5cc2112 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/16023.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $391^m \equiv 1 \pmod{655}$. +Answer: +$65$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/17927.txt b/pretraining/mathematica/number_theory/multiplicative_order/17927.txt new file mode 100644 index 0000000000000000000000000000000000000000..a959f0c16a1f83f8213afa19c38a45b6f27c0e8f --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/17927.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $212^m \equiv 1 \pmod{721}$. +Answer: +$102$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/17959.txt b/pretraining/mathematica/number_theory/multiplicative_order/17959.txt new file mode 100644 index 0000000000000000000000000000000000000000..652d7601964a1fca985cf8d898b1592a2bf7bbf5 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/17959.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $262^m \equiv 1 \pmod{867}$. +Answer: +$272$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/19243.txt b/pretraining/mathematica/number_theory/multiplicative_order/19243.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5f7ce450da622437544c2f20bbb74073a4afb30 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/19243.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $311^m \equiv 1 \pmod{826}$. +Answer: +$174$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/21365.txt b/pretraining/mathematica/number_theory/multiplicative_order/21365.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e6eda2a202e3576e2ffc432203e155ec46ddef0 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/21365.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $325^m \equiv 1 \pmod{682}$. +Answer: +$10$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/23015.txt b/pretraining/mathematica/number_theory/multiplicative_order/23015.txt new file mode 100644 index 0000000000000000000000000000000000000000..6792ba8d8c8801b0770da688a8c97a5749b76eb8 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/23015.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $248^m \equiv 1 \pmod{349}$. +Answer: +$116$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/25071.txt b/pretraining/mathematica/number_theory/multiplicative_order/25071.txt new file mode 100644 index 0000000000000000000000000000000000000000..544ec8baeb52e079bb89719be533dd5fc8e3bb77 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/25071.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $669^m \equiv 1 \pmod{778}$. +Answer: +$388$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/27174.txt b/pretraining/mathematica/number_theory/multiplicative_order/27174.txt new file mode 100644 index 0000000000000000000000000000000000000000..3baa94eb18482b42793e2587eda09cb38c2ecf0f --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/27174.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $111^m \equiv 1 \pmod{986}$. +Answer: +$56$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/28293.txt b/pretraining/mathematica/number_theory/multiplicative_order/28293.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ad5588bd17aed748b484f3e00f4ed8ee6764a2d --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/28293.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $109^m \equiv 1 \pmod{230}$. +Answer: +$22$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/28825.txt b/pretraining/mathematica/number_theory/multiplicative_order/28825.txt new file mode 100644 index 0000000000000000000000000000000000000000..655e28b5026f62db90d55de9efa9f7127d6e09ec --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/28825.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $16^m \equiv 1 \pmod{61}$. +Answer: +$15$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/29035.txt b/pretraining/mathematica/number_theory/multiplicative_order/29035.txt new file mode 100644 index 0000000000000000000000000000000000000000..402422dbc2eb0d58d902357c1056d52d32b39613 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/29035.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $371^m \equiv 1 \pmod{451}$. +Answer: +$20$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/30271.txt b/pretraining/mathematica/number_theory/multiplicative_order/30271.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d742d8dbd8ccab630f2b7f51e3edea84b042ba --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/30271.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $186^m \equiv 1 \pmod{449}$. +Answer: +$224$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/30632.txt b/pretraining/mathematica/number_theory/multiplicative_order/30632.txt new file mode 100644 index 0000000000000000000000000000000000000000..134d1bc4c72b314c19119207e296a1930ed7c84e --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/30632.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $74^m \equiv 1 \pmod{165}$. +Answer: +$10$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/31561.txt b/pretraining/mathematica/number_theory/multiplicative_order/31561.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9dfcaae310bff2d782883f2eb770a79e168097c --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/31561.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $5^m \equiv 1 \pmod{396}$. +Answer: +$30$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/33413.txt b/pretraining/mathematica/number_theory/multiplicative_order/33413.txt new file mode 100644 index 0000000000000000000000000000000000000000..1cd5f9b9d855cf2be61b6c3df5b03d74bf619eba --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/33413.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $666^m \equiv 1 \pmod{749}$. +Answer: +$106$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/34091.txt b/pretraining/mathematica/number_theory/multiplicative_order/34091.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3779e3574899f536ebb6bed470a00675c16cb7f --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/34091.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $385^m \equiv 1 \pmod{642}$. +Answer: +$53$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/34295.txt b/pretraining/mathematica/number_theory/multiplicative_order/34295.txt new file mode 100644 index 0000000000000000000000000000000000000000..9166572255683674fbd252084417a3b2d6c6bc67 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/34295.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $397^m \equiv 1 \pmod{488}$. +Answer: +$60$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/35828.txt b/pretraining/mathematica/number_theory/multiplicative_order/35828.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b67d6d44f1541b29c9c666354d756ceb1d50b67 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/35828.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $167^m \equiv 1 \pmod{887}$. +Answer: +$443$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/36122.txt b/pretraining/mathematica/number_theory/multiplicative_order/36122.txt new file mode 100644 index 0000000000000000000000000000000000000000..e057e737c61306685be3981776eec7bfb996d44e --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/36122.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $52^m \equiv 1 \pmod{651}$. +Answer: +$30$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/36203.txt b/pretraining/mathematica/number_theory/multiplicative_order/36203.txt new file mode 100644 index 0000000000000000000000000000000000000000..248beddff2dd75a43cdb6d5b52fe0c835e043ec0 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/36203.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $446^m \equiv 1 \pmod{575}$. +Answer: +$55$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/4067.txt b/pretraining/mathematica/number_theory/multiplicative_order/4067.txt new file mode 100644 index 0000000000000000000000000000000000000000..017b9b946347e4dd31b08ac569162d073362a7a6 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/4067.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $50^m \equiv 1 \pmod{517}$. +Answer: +$230$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/40879.txt b/pretraining/mathematica/number_theory/multiplicative_order/40879.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c4a5f534e0090337d342469a76e70ab4f821f5e --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/40879.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $368^m \equiv 1 \pmod{473}$. +Answer: +$105$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/41316.txt b/pretraining/mathematica/number_theory/multiplicative_order/41316.txt new file mode 100644 index 0000000000000000000000000000000000000000..733a40a3b42deb11954daf1f9e1d6f1860a89891 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/41316.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $73^m \equiv 1 \pmod{228}$. +Answer: +$9$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/41600.txt b/pretraining/mathematica/number_theory/multiplicative_order/41600.txt new file mode 100644 index 0000000000000000000000000000000000000000..9885d1efb9494c3b3fadd29d7c1b127a7a2ff5f5 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/41600.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $95^m \equiv 1 \pmod{99}$. +Answer: +$30$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/44504.txt b/pretraining/mathematica/number_theory/multiplicative_order/44504.txt new file mode 100644 index 0000000000000000000000000000000000000000..bac35748b680a6c17e88dbc44d583038af1a12b0 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/44504.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $49^m \equiv 1 \pmod{254}$. +Answer: +$63$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/44673.txt b/pretraining/mathematica/number_theory/multiplicative_order/44673.txt new file mode 100644 index 0000000000000000000000000000000000000000..798a9647c6d31124be7875ce4f3a51f7c97447b2 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/44673.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $581^m \equiv 1 \pmod{748}$. +Answer: +$80$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/44807.txt b/pretraining/mathematica/number_theory/multiplicative_order/44807.txt new file mode 100644 index 0000000000000000000000000000000000000000..6eccd78c82d7bb26d378e9e169d70eef794418e9 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/44807.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $791^m \equiv 1 \pmod{828}$. +Answer: +$22$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/46353.txt b/pretraining/mathematica/number_theory/multiplicative_order/46353.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d74aee698a4027dc853c7b53b3b171561b973ec --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/46353.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $483^m \equiv 1 \pmod{775}$. +Answer: +$60$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/46503.txt b/pretraining/mathematica/number_theory/multiplicative_order/46503.txt new file mode 100644 index 0000000000000000000000000000000000000000..31cec120bfef3c36088149a59a45ad964359df99 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/46503.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $545^m \equiv 1 \pmod{624}$. +Answer: +$2$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/48363.txt b/pretraining/mathematica/number_theory/multiplicative_order/48363.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1ec4743df1caf17f499b74dcdf610f77bd28b1c --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/48363.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $882^m \equiv 1 \pmod{985}$. +Answer: +$196$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/48680.txt b/pretraining/mathematica/number_theory/multiplicative_order/48680.txt new file mode 100644 index 0000000000000000000000000000000000000000..6cb34150d842fb1169c694306dcd318a375bd55a --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/48680.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $100^m \equiv 1 \pmod{231}$. +Answer: +$3$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/6995.txt b/pretraining/mathematica/number_theory/multiplicative_order/6995.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d759440d49d86e1693be7422b0e23f36fd5fc56 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/6995.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $15^m \equiv 1 \pmod{629}$. +Answer: +$72$ \ No newline at end of file diff --git a/pretraining/mathematica/number_theory/multiplicative_order/7607.txt b/pretraining/mathematica/number_theory/multiplicative_order/7607.txt new file mode 100644 index 0000000000000000000000000000000000000000..4bae419135e45aead493d1f32ef4ba48cf416bc8 --- /dev/null +++ b/pretraining/mathematica/number_theory/multiplicative_order/7607.txt @@ -0,0 +1,4 @@ +Problem: +Find the smallest integer $m$ such that $57^m \equiv 1 \pmod{871}$. +Answer: +$132$ \ No newline at end of file