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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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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}

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:
+$33$

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:
+$316$

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:
+$18$

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:
+$132$

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:
+$18$

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:
+$65$

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:
+$102$

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:
+$272$

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:
+$174$

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:
+$10$

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:
+$116$

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:
+$388$

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:
+$56$