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pretraining/mathematica/linear_algebra/multiply_w_steps/1024.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ \frac{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ \frac{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{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 }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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{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: }\frac{12\ 16}{7\ 7}+2 (-2)=-\frac{4}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{4}{49}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{12\ 3}{7\ 7}+\frac{2\ 3}{7}=\frac{78}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \fbox{$\frac{78}{49}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{12\ 15}{7\ 7}+\frac{2\ 9}{7}=\frac{306}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \fbox{$\frac{306}{49}$} \\
+ \_ & \_ & \_ \\
+\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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{7} \left(-\frac{16}{7}\right)+\left(-\frac{10}{7}\right)\, (-2)=\frac{124}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \fbox{$\frac{124}{49}$} & \_ & \_ \\
+\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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \frac{124}{49} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{7} \left(-\frac{3}{7}\right)+\left(-\frac{10}{7}\right)\, \times \, \frac{3}{7}=-\frac{33}{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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \frac{124}{49} & \fbox{$-\frac{33}{49}$} & \_ \\
+\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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \frac{124}{49} & -\frac{33}{49} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{7} \left(-\frac{15}{7}\right)+\left(-\frac{10}{7}\right)\, \times \, \frac{9}{7}=-\frac{15}{7}. \\
+ \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{12}{7} & 2 \\
+ -\frac{1}{7} & -\frac{10}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{16}{7} & \frac{3}{7} & \frac{15}{7} \\
+ -2 & \frac{3}{7} & \frac{9}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{49} & \frac{78}{49} & \frac{306}{49} \\
+ \frac{124}{49} & -\frac{33}{49} & \fbox{$-\frac{15}{7}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1056.txt

@@ -0,0 +1,549 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -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 }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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 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{1}{4} \left(-\frac{5}{2}\right)+\frac{1}{4} \left(-\frac{1}{4}\right)-\frac{7}{2\ 4}=-\frac{25}{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}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{25}{16}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\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{1}{4} (-1)+\frac{1}{4} \left(-\frac{1}{2}\right)-\frac{1}{2}=-\frac{7}{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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & \fbox{$-\frac{7}{8}$} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\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{1}{4} \left(-\frac{7}{4}\right)+\frac{2}{4}+\frac{3}{2}=\frac{25}{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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \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}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{2}\right)\, \times \, \frac{5}{2}+\frac{1}{4\ 4}+\frac{9 (-7)}{4\ 4}=-\frac{61}{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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ \fbox{$-\frac{61}{8}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{2}\right)\, \times \, 1+\frac{1}{4\ 2}+\frac{9 (-1)}{4}=-\frac{29}{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}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & \fbox{$-\frac{29}{8}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{2}\right)\, \times \, \frac{7}{4}-\frac{2}{4}+\frac{9\ 3}{4}=\frac{29}{8}. \\
+ \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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \fbox{$\frac{29}{8}$} \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3\ 5}{2\ 2}+\left(-\frac{7}{4}\right)\, \times \, \frac{1}{4}+\frac{9 (-7)}{4\ 4}=-\frac{5}{8}. \\
+ \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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ \fbox{$-\frac{5}{8}$} & \_ & \_ \\
+\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{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ -\frac{5}{8} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3}{2}+\left(-\frac{7}{4}\right)\, \times \, \frac{1}{2}+\frac{9 (-1)}{4}=-\frac{13}{8}. \\
+ \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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ -\frac{5}{8} & \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 }3^{\text{rd}} \text{column}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ -\frac{5}{8} & -\frac{13}{8} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3\ 7}{2\ 4}+\left(-\frac{7}{4}\right)\, (-2)+\frac{9\ 3}{4}=\frac{103}{8}. \\
+ \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}
+ -\frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\
+ -\frac{3}{2} & \frac{1}{4} & \frac{9}{4} \\
+ \frac{3}{2} & -\frac{7}{4} & \frac{9}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{5}{2} & 1 & \frac{7}{4} \\
+ \frac{1}{4} & \frac{1}{2} & -2 \\
+ -\frac{7}{4} & -1 & 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{25}{16} & -\frac{7}{8} & \frac{25}{16} \\
+ -\frac{61}{8} & -\frac{29}{8} & \frac{29}{8} \\
+ -\frac{5}{8} & -\frac{13}{8} & \fbox{$\frac{103}{8}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1059.txt

@@ -0,0 +1,166 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{8} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{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 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 }2\times 1: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{ccc}
+ \frac{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{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}{ccc}
+ \frac{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{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{7\ 17}{8\ 8}+\left(-\frac{15}{8}\right)\, \left(-\frac{11}{8}\right)+\left(-\frac{5}{2}\right)\, \left(-\frac{7}{8}\right)=\frac{53}{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}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$\frac{53}{8}$} \\
+ \_ \\
+\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{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{53}{8} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 17}{8}+(-1)\, \left(-\frac{11}{8}\right)+\left(-\frac{13}{8}\right)\, \left(-\frac{7}{8}\right)=\frac{451}{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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ \frac{7}{8} & -\frac{15}{8} & -\frac{5}{2} \\
+ 2 & -1 & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{17}{8} \\
+ -\frac{11}{8} \\
+ -\frac{7}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{53}{8} \\
+ \fbox{$\frac{451}{64}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1179.txt

@@ -0,0 +1,549 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{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 }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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\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 (-13)}{5\ 5}+\frac{9 (-13)}{5\ 5}+\frac{13 (-2)}{5\ 5}=-\frac{39}{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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{39}{5}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4\ 3}{5}+\frac{9 (-8)}{5\ 5}+\frac{13 (-2)}{5}=-\frac{142}{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}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & \fbox{$-\frac{142}{25}$} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4 (-11)}{5\ 5}+\frac{9}{5}+\frac{13 (-13)}{5\ 5}=-\frac{168}{25}. \\
+ \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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & \fbox{$-\frac{168}{25}$} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7 (-13)}{5\ 5}-\frac{13}{5}+\left(-\frac{2}{5}\right)\, \left(-\frac{2}{5}\right)=-\frac{152}{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}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ \fbox{$-\frac{152}{25}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7\ 3}{5}-\frac{8}{5}+\left(-\frac{2}{5}\right)\, (-2)=\frac{17}{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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \fbox{$\frac{17}{5}$} & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7 (-11)}{5\ 5}+1\ 1+\left(-\frac{2}{5}\right)\, \left(-\frac{13}{5}\right)=-\frac{26}{25}. \\
+ \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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & \fbox{$-\frac{26}{25}$} \\
+ \_ & \_ & \_ \\
+\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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13 (-13)}{5\ 5}+(-3)\, \left(-\frac{13}{5}\right)+\left(-\frac{13}{5}\right)\, \left(-\frac{2}{5}\right)=\frac{52}{25}. \\
+ \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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \fbox{$\frac{52}{25}$} & \_ & \_ \\
+\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{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \frac{52}{25} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13\ 3}{5}+(-3)\, \left(-\frac{8}{5}\right)+\left(-\frac{13}{5}\right)\, (-2)=\frac{89}{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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \frac{52}{25} & \fbox{$\frac{89}{5}$} & \_ \\
+\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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \frac{52}{25} & \frac{89}{5} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{13 (-11)}{5\ 5}+(-3)\, \times \, 1+\left(-\frac{13}{5}\right)\, \left(-\frac{13}{5}\right)=-\frac{49}{25}. \\
+ \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}
+ \frac{4}{5} & \frac{9}{5} & \frac{13}{5} \\
+ \frac{7}{5} & 1 & -\frac{2}{5} \\
+ \frac{13}{5} & -3 & -\frac{13}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & 3 & -\frac{11}{5} \\
+ -\frac{13}{5} & -\frac{8}{5} & 1 \\
+ -\frac{2}{5} & -2 & -\frac{13}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{39}{5} & -\frac{142}{25} & -\frac{168}{25} \\
+ -\frac{152}{25} & \frac{17}{5} & -\frac{26}{25} \\
+ \frac{52}{25} & \frac{89}{5} & \fbox{$-\frac{49}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1180.txt

@@ -0,0 +1,166 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -1 \\
+ 0 \\
+ -2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 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 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 }2\times 1: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{ccc}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 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 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 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)\, (-1)+0\ 0+0 (-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}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 0 \\
+ -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}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 0 \\
+ -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: }(-1)\, (-1)+1\ 0+(-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ -3 & 0 & 0 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ 0 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 3 \\
+ \fbox{$3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1228.txt

@@ -0,0 +1,231 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{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 }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{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\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{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\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{7}{5}\right)\, \left(-\frac{12}{5}\right)+\frac{2\ 7}{5\ 5}+\left(-\frac{14}{5}\right)\, \left(-\frac{6}{5}\right)=\frac{182}{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}{ccc}
+ -\frac{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$\frac{182}{25}$} \\
+ \_ \\
+ \_ \\
+\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}{ccc}
+ -\frac{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{182}{25} \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{5}\right)\, \left(-\frac{12}{5}\right)+\left(-\frac{11}{5}\right)\, \times \, \frac{7}{5}+\left(-\frac{9}{5}\right)\, \left(-\frac{6}{5}\right)=\frac{17}{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}
+ \\
+ \left(
+\begin{array}{ccc}
+ -\frac{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{182}{25} \\
+ \fbox{$\frac{17}{5}$} \\
+ \_ \\
+\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{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{182}{25} \\
+ \frac{17}{5} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{5}\right)\, \left(-\frac{12}{5}\right)+\frac{3\ 7}{5\ 5}+\frac{9 (-6)}{5\ 5}=\frac{51}{25}. \\
+ \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{7}{5} & \frac{2}{5} & -\frac{14}{5} \\
+ -\frac{9}{5} & -\frac{11}{5} & -\frac{9}{5} \\
+ -\frac{7}{5} & \frac{3}{5} & \frac{9}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -\frac{12}{5} \\
+ \frac{7}{5} \\
+ -\frac{6}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{182}{25} \\
+ \frac{17}{5} \\
+ \fbox{$\frac{51}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/124.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 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}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 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}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 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: }(-1)\, (-2)+(-1)\, \times \, 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 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\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}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\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: }(-1)\, (-2)+(-1)\, \times \, 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}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & \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}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & \_ \\
+ \_ & \_ & \_ \\
+\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)\, \times \, 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}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & \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}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & -1 \\
+ \_ & \_ & \_ \\
+\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=-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 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & -1 \\
+ \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 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 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 (-2)+(-1)\, \times \, 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}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & -1 \\
+ -4 & \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}{cc}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & -1 \\
+ -4 & -5 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 0+(-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}
+ -1 & -1 \\
+ 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -2 & -2 & 0 \\
+ 0 & 1 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & 1 & -1 \\
+ -4 & -5 & \fbox{$-1$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1241.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -\frac{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -\frac{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{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 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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\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{12}{5}\right)\, \times \, \frac{12}{5}+\frac{13 (-3)}{5}=-\frac{339}{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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{339}{25}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{12}{5}\right)\, (-3)+\frac{13\ 3}{5}=15. \\
+ \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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & \fbox{$15$} & \_ \\
+ \_ & \_ & \_ \\
+\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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{12}{5}\right)\, (-2)+\frac{13 (-11)}{5\ 5}=-\frac{23}{25}. \\
+ \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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & \fbox{$-\frac{23}{25}$} \\
+ \_ & \_ & \_ \\
+\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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{5}\right)\, \times \, \frac{12}{5}+\left(-\frac{4}{5}\right)\, (-3)=-\frac{12}{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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ \fbox{$-\frac{12}{25}$} & \_ & \_ \\
+\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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ -\frac{12}{25} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{5}\right)\, (-3)+\left(-\frac{4}{5}\right)\, \times \, 3=\frac{6}{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}{cc}
+ -\frac{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ -\frac{12}{25} & \fbox{$\frac{6}{5}$} & \_ \\
+\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}{cc}
+ -\frac{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ -\frac{12}{25} & \frac{6}{5} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{5}\right)\, (-2)+\left(-\frac{4}{5}\right)\, \left(-\frac{11}{5}\right)=\frac{104}{25}. \\
+ \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{12}{5} & \frac{13}{5} \\
+ -\frac{6}{5} & -\frac{4}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{12}{5} & -3 & -2 \\
+ -3 & 3 & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{339}{25} & 15 & -\frac{23}{25} \\
+ -\frac{12}{25} & \frac{6}{5} & \fbox{$\frac{104}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/125.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{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 }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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{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: }\left(-\frac{13}{5}\right)\, \times \, \frac{14}{5}+\left(-\frac{8}{5}\right)\, \times \, 1+\frac{7 (-4)}{5\ 5}=-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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$-10$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{13}{5}\right)\, \left(-\frac{7}{5}\right)+\left(-\frac{8}{5}\right)\, \times \, 1+\frac{7 (-11)}{5\ 5}=-\frac{26}{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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & \fbox{$-\frac{26}{25}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{2}{5}\right)\, \times \, \frac{14}{5}+\frac{6}{5}+\frac{4}{5\ 5}=\frac{6}{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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \fbox{$\frac{6}{25}$} & \_ \\
+ \_ & \_ \\
+\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}{ccc}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{2}{5}\right)\, \left(-\frac{7}{5}\right)+\frac{6}{5}+\frac{11}{5\ 5}=\frac{11}{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}
+ -\frac{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \fbox{$\frac{11}{5}$} \\
+ \_ & \_ \\
+\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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \frac{11}{5} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{5}\right)\, \times \, \frac{14}{5}+\frac{14}{5}+\left(-\frac{11}{5}\right)\, \left(-\frac{4}{5}\right)=\frac{6}{5}. \\
+ \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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \frac{11}{5} \\
+ \fbox{$\frac{6}{5}$} & \_ \\
+\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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \frac{11}{5} \\
+ \frac{6}{5} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{6}{5}\right)\, \left(-\frac{7}{5}\right)+\frac{14}{5}+\left(-\frac{11}{5}\right)\, \left(-\frac{11}{5}\right)=\frac{233}{25}. \\
+ \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{13}{5} & -\frac{8}{5} & \frac{7}{5} \\
+ -\frac{2}{5} & \frac{6}{5} & -\frac{1}{5} \\
+ -\frac{6}{5} & \frac{14}{5} & -\frac{11}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{14}{5} & -\frac{7}{5} \\
+ 1 & 1 \\
+ -\frac{4}{5} & -\frac{11}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -10 & -\frac{26}{25} \\
+ \frac{6}{25} & \frac{11}{5} \\
+ \frac{6}{5} & \fbox{$\frac{233}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/127.txt

@@ -0,0 +1,231 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 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 }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}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -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}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -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: }0 (-3)+(-1)\, \times \, 1+(-2)\, (-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}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-3)+1\ 1+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}
+ \\
+ \left(
+\begin{array}{ccc}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ \fbox{$5$} \\
+ \_ \\
+\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 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ 5 \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, (-3)+(-3)\, \times \, 1+3 (-1)=-3. \\
+ \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}
+ 0 & -1 & -2 \\
+ -2 & 1 & 2 \\
+ -1 & -3 & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 1 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ 5 \\
+ \fbox{$-3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/132.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\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 }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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\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}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\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{2}{2}+\frac{9 (-3)}{4\ 4}=-\frac{11}{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}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{11}{16}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, \frac{1}{2}+\frac{9\ 2}{4}=\frac{7}{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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \fbox{$\frac{7}{2}$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 0+\frac{9 (-3)}{4\ 2}=-\frac{27}{8}. \\
+ \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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & \fbox{$-\frac{27}{8}$} \\
+ \_ & \_ & \_ \\
+\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}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{2} \left(-\frac{3}{4}\right)+\frac{5 (-3)}{4\ 4}=-\frac{21}{16}. \\
+ \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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ \fbox{$-\frac{21}{16}$} & \_ & \_ \\
+\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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ -\frac{21}{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}+\frac{5\ 2}{4}=\frac{23}{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}
+ -2 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ -\frac{21}{16} & \fbox{$\frac{23}{8}$} & \_ \\
+\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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ -\frac{21}{16} & \frac{23}{8} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3\ 0}{4}+\frac{5 (-3)}{4\ 2}=-\frac{15}{8}. \\
+ \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 & \frac{9}{4} \\
+ \frac{3}{4} & \frac{5}{4} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & \frac{1}{2} & 0 \\
+ -\frac{3}{4} & 2 & -\frac{3}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{11}{16} & \frac{7}{2} & -\frac{27}{8} \\
+ -\frac{21}{16} & \frac{23}{8} & \fbox{$-\frac{15}{8}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1364.txt

@@ -0,0 +1,264 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ 3 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ 3 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 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 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}
+ 3 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 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}
+ 3 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 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: }3\ 1+2\ 3+3\ 3=18. \\
+ \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 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$18$} & \_ \\
+ \_ & \_ \\
+\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 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3 (-2)+2 (-3)+3\ 1=-9. \\
+ \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 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & \fbox{$-9$} \\
+ \_ & \_ \\
+\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 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & -9 \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 1+(-2)\, \times \, 3+(-2)\, \times \, 3=-15. \\
+ \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 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & -9 \\
+ \fbox{$-15$} & \_ \\
+\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 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & -9 \\
+ -15 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-2)+(-2)\, (-3)+(-2)\, \times \, 1=10. \\
+ \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}
+ 3 & 2 & 3 \\
+ -3 & -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 3 & -3 \\
+ 3 & 1 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 18 & -9 \\
+ -15 & \fbox{$10$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1405.txt

@@ -0,0 +1,231 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{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 }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}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\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}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\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{5}{4\ 8}+\frac{11}{8\ 8}+\frac{1}{8} \left(-\frac{43}{16}\right)=-\frac{1}{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{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$-\frac{1}{128}$} \\
+ \_ \\
+ \_ \\
+\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{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{1}{128} \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{13}{8}\right)\, \times \, \frac{1}{8}+\frac{1}{16} \left(-\frac{11}{8}\right)+\left(-\frac{21}{16}\right)\, \times \, \frac{43}{16}=-\frac{977}{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{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{1}{128} \\
+ \fbox{$-\frac{977}{256}$} \\
+ \_ \\
+\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{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{1}{128} \\
+ -\frac{977}{256} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{31}{16}\right)\, \times \, \frac{1}{8}+\left(-\frac{13}{16}\right)\, \times \, \frac{11}{8}+\left(-\frac{33}{16}\right)\, \times \, \frac{43}{16}=-\frac{1767}{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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ \frac{5}{4} & \frac{1}{8} & -\frac{1}{8} \\
+ -\frac{13}{8} & -\frac{1}{16} & -\frac{21}{16} \\
+ -\frac{31}{16} & -\frac{13}{16} & -\frac{33}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{1}{8} \\
+ \frac{11}{8} \\
+ \frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{1}{128} \\
+ -\frac{977}{256} \\
+ \fbox{$-\frac{1767}{256}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1426.txt

@@ -0,0 +1,362 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{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 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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\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{19}{8}\right)\, \left(-\frac{7}{8}\right)+\frac{2 (-11)}{4}+\frac{3\ 0}{8}=-\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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\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: }\left(-\frac{19}{8}\right)\, \times \, \frac{7}{4}+\frac{2 (-11)}{16}+\frac{3 (-27)}{8\ 16}=-\frac{789}{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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & \fbox{$-\frac{789}{128}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{19}{8}\right)\, \times \, \frac{11}{16}+\frac{2\ 7}{4}+\frac{3\ 17}{8\ 8}=\frac{341}{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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \fbox{$\frac{341}{128}$} \\
+ \_ & \_ & \_ \\
+\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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7}{8\ 2}+\frac{37 (-11)}{16\ 4}+\left(-\frac{21}{16}\right)\, \times \, 0=-\frac{379}{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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ \fbox{$-\frac{379}{64}$} & \_ & \_ \\
+\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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ -\frac{379}{64} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{2} \left(-\frac{7}{4}\right)+\frac{37 (-11)}{16\ 16}+\left(-\frac{21}{16}\right)\, \left(-\frac{27}{16}\right)=-\frac{1}{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{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ -\frac{379}{64} & \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 }3^{\text{rd}} \text{column}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ -\frac{379}{64} & -\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}{2} \left(-\frac{11}{16}\right)+\frac{37\ 7}{16\ 4}+\left(-\frac{21}{16}\right)\, \times \, \frac{17}{8}=\frac{117}{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}{ccc}
+ -\frac{19}{8} & 2 & \frac{3}{8} \\
+ -\frac{1}{2} & \frac{37}{16} & -\frac{21}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{7}{8} & \frac{7}{4} & \frac{11}{16} \\
+ -\frac{11}{4} & -\frac{11}{16} & \frac{7}{4} \\
+ 0 & -\frac{27}{16} & \frac{17}{8} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{219}{64} & -\frac{789}{128} & \frac{341}{128} \\
+ -\frac{379}{64} & -\frac{1}{4} & \fbox{$\frac{117}{128}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1484.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -2 \\
+ -1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -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 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}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -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}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -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: }0 (-2)+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}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -1 \\
+ \_ \\
+\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)=-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ 0 & 1 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -1 \\
+ \fbox{$-4$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1489.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 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 }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 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 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}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 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: }(-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 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\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}{cc}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\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: }(-2)\, \times \, 1+0\ 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 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & \fbox{$-2$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -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 }3^{\text{rd}} \text{column }\text{of }\text{the }\text{product}: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{cc}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -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 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -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\ 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}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -2 & 4 \\
+ \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}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -2 & 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+2\ 3=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}
+ -2 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -2 & 4 \\
+ 2 & \fbox{$8$} & \_ \\
+\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 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -2 & 4 \\
+ 2 & 8 & \_ \\
+\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. \\
+ \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 & 0 \\
+ 2 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & 1 & -2 \\
+ 2 & 3 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 2 & -2 & 4 \\
+ 2 & 8 & \fbox{$0$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/152.txt

@@ -0,0 +1,166 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{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 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 }2\times 1: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\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{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\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{21\ 12}{10\ 5}+\frac{3}{5\ 5}+\frac{3 (-12)}{5\ 5}=\frac{93}{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}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$\frac{93}{25}$} \\
+ \_ \\
+\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}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{93}{25} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{19}{10}\right)\, \times \, \frac{12}{5}+\frac{1}{5} \left(-\frac{17}{10}\right)+\left(-\frac{21}{10}\right)\, \left(-\frac{12}{5}\right)=\frac{7}{50}. \\
+ \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}{ccc}
+ \frac{21}{10} & -\frac{3}{5} & \frac{3}{5} \\
+ -\frac{19}{10} & \frac{17}{10} & -\frac{21}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{12}{5} \\
+ -\frac{1}{5} \\
+ -\frac{12}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{93}{25} \\
+ \fbox{$\frac{7}{50}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1628.txt

@@ -0,0 +1,362 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 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 }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}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 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)\, \left(-\frac{5}{3}\right)+0\ 2+\left(-\frac{7}{3}\right)\, \times \, 0=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}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 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)\, (-2)+\frac{0 (-4)}{3}+\left(-\frac{7}{3}\right)\, \times \, 2=\frac{4}{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}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \fbox{$\frac{4}{3}$} & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, \frac{2}{3}+\frac{0 (-4)}{3}+\left(-\frac{7}{3}\right)\, (-1)=\frac{1}{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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \fbox{$\frac{1}{3}$} \\
+ \_ & \_ & \_ \\
+\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}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5 (-5)}{3\ 3}+\frac{5\ 2}{3}+\left(-\frac{8}{3}\right)\, \times \, 0=\frac{5}{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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \fbox{$\frac{5}{9}$} & \_ & \_ \\
+\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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \frac{5}{9} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5 (-2)}{3}+\frac{5 (-4)}{3\ 3}+\left(-\frac{8}{3}\right)\, \times \, 2=-\frac{98}{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}
+ \\
+ \left(
+\begin{array}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \frac{5}{9} & \fbox{$-\frac{98}{9}$} & \_ \\
+\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 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \frac{5}{9} & -\frac{98}{9} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5\ 2}{3\ 3}+\frac{5 (-4)}{3\ 3}+\left(-\frac{8}{3}\right)\, (-1)=\frac{14}{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}{ccc}
+ -3 & 0 & -\frac{7}{3} \\
+ \frac{5}{3} & \frac{5}{3} & -\frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{5}{3} & -2 & \frac{2}{3} \\
+ 2 & -\frac{4}{3} & -\frac{4}{3} \\
+ 0 & 2 & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 5 & \frac{4}{3} & \frac{1}{3} \\
+ \frac{5}{9} & -\frac{98}{9} & \fbox{$\frac{14}{9}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1784.txt

@@ -0,0 +1,375 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -\frac{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -\frac{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{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 }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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -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}{5}\right)\, \times \, \frac{2}{5}+\left(-\frac{14}{5}\right)\, \left(-\frac{3}{5}\right)=\frac{24}{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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$\frac{24}{25}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{5}\right)\, \left(-\frac{11}{5}\right)+\left(-\frac{14}{5}\right)\, (-2)=\frac{239}{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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \fbox{$\frac{239}{25}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{14}{5}\right)\, \times \, \frac{2}{5}+\frac{6 (-3)}{5\ 5}=-\frac{46}{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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ \fbox{$-\frac{46}{25}$} & \_ \\
+ \_ & \_ \\
+\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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{14}{5}\right)\, \left(-\frac{11}{5}\right)+\frac{6 (-2)}{5}=\frac{94}{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}
+ \\
+ \left(
+\begin{array}{cc}
+ -\frac{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \fbox{$\frac{94}{25}$} \\
+ \_ & \_ \\
+\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}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \frac{94}{25} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{5}\right)\, \times \, \frac{2}{5}+\frac{8 (-3)}{5\ 5}=-\frac{6}{5}. \\
+ \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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \frac{94}{25} \\
+ \fbox{$-\frac{6}{5}$} & \_ \\
+\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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \frac{94}{25} \\
+ -\frac{6}{5} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{5}\right)\, \left(-\frac{11}{5}\right)+\frac{8 (-2)}{5}=-\frac{47}{25}. \\
+ \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{9}{5} & -\frac{14}{5} \\
+ -\frac{14}{5} & \frac{6}{5} \\
+ -\frac{3}{5} & \frac{8}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{2}{5} & -\frac{11}{5} \\
+ -\frac{3}{5} & -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{24}{25} & \frac{239}{25} \\
+ -\frac{46}{25} & \frac{94}{25} \\
+ -\frac{6}{5} & \fbox{$-\frac{47}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1787.txt

@@ -0,0 +1,528 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{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 }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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{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: }\left(-\frac{7}{8}\right)\, \times \, \frac{11}{4}+\frac{23\ 19}{8\ 8}=\frac{283}{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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$\frac{283}{64}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{8}\right)\, \times \, \frac{15}{8}+\frac{23 (-11)}{8\ 16}=-\frac{463}{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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & \fbox{$-\frac{463}{128}$} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{7}{8}\right)\, \times \, \frac{37}{16}+\frac{23 (-43)}{8\ 16}=-\frac{39}{4}. \\
+ \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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & \fbox{$-\frac{39}{4}$} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{11}{4}+\left(-\frac{31}{16}\right)\, \times \, \frac{19}{8}=-\frac{1469}{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}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ \fbox{$-\frac{1469}{128}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{15}{8}+\left(-\frac{31}{16}\right)\, \left(-\frac{11}{16}\right)=-\frac{859}{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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & \fbox{$-\frac{859}{256}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{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)\, \times \, \frac{37}{16}+\left(-\frac{31}{16}\right)\, \left(-\frac{43}{16}\right)=-\frac{147}{256}. \\
+ \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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & \fbox{$-\frac{147}{256}$} \\
+ \_ & \_ & \_ \\
+\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}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{8} \left(-\frac{11}{4}\right)+\left(-\frac{13}{8}\right)\, \times \, \frac{19}{8}=-\frac{269}{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}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ \fbox{$-\frac{269}{64}$} & \_ & \_ \\
+\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}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ -\frac{269}{64} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{8} \left(-\frac{15}{8}\right)+\left(-\frac{13}{8}\right)\, \left(-\frac{11}{16}\right)=\frac{113}{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}
+ \\
+ \left(
+\begin{array}{cc}
+ -\frac{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ -\frac{269}{64} & \fbox{$\frac{113}{128}$} & \_ \\
+\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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ -\frac{269}{64} & \frac{113}{128} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{8} \left(-\frac{37}{16}\right)+\left(-\frac{13}{8}\right)\, \left(-\frac{43}{16}\right)=\frac{261}{64}. \\
+ \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{7}{8} & \frac{23}{8} \\
+ -\frac{5}{2} & -\frac{31}{16} \\
+ -\frac{1}{8} & -\frac{13}{8} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{11}{4} & \frac{15}{8} & \frac{37}{16} \\
+ \frac{19}{8} & -\frac{11}{16} & -\frac{43}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{283}{64} & -\frac{463}{128} & -\frac{39}{4} \\
+ -\frac{1469}{128} & -\frac{859}{256} & -\frac{147}{256} \\
+ -\frac{269}{64} & \frac{113}{128} & \fbox{$\frac{261}{64}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1795.txt

@@ -0,0 +1,549 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\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{1}{6} (-1)+\frac{3}{3\ 2}+(-1)\, (-1)=\frac{4}{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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$\frac{4}{3}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }-\frac{5}{2}+\frac{0}{3}+(-1)\, (-2)=-\frac{1}{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}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & \fbox{$-\frac{1}{2}$} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-2)+\frac{5}{3\ 2}+(-1)\, \left(-\frac{11}{6}\right)=\frac{2}{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}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \fbox{$\frac{2}{3}$} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \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{2}{6}+\frac{0\ 3}{2}+\left(-\frac{5}{6}\right)\, (-1)=\frac{7}{6}. \\
+ \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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \fbox{$\frac{7}{6}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \left(-\frac{5}{2}\right)+0\ 0+\left(-\frac{5}{6}\right)\, (-2)=\frac{20}{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}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \fbox{$\frac{20}{3}$} & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+\frac{0\ 5}{2}+\left(-\frac{5}{6}\right)\, \left(-\frac{11}{6}\right)=\frac{199}{36}. \\
+ \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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \fbox{$\frac{199}{36}$} \\
+ \_ & \_ & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{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} \left(-\frac{5}{2}\right)+\frac{5\ 3}{6\ 2}+\frac{1}{6}=1. \\
+ \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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{36} \\
+ \fbox{$1$} & \_ & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{36} \\
+ 1 & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5 (-5)}{2\ 2}+\frac{5\ 0}{6}+\frac{2}{6}=-\frac{71}{12}. \\
+ \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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{36} \\
+ 1 & \fbox{$-\frac{71}{12}$} & \_ \\
+\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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{36} \\
+ 1 & -\frac{71}{12} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5 (-2)}{2}+\frac{5\ 5}{6\ 2}+\frac{11}{6\ 6}=-\frac{47}{18}. \\
+ \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}
+ 1 & \frac{1}{3} & -1 \\
+ -2 & 0 & -\frac{5}{6} \\
+ \frac{5}{2} & \frac{5}{6} & -\frac{1}{6} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{6} & -\frac{5}{2} & -2 \\
+ \frac{3}{2} & 0 & \frac{5}{2} \\
+ -1 & -2 & -\frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{4}{3} & -\frac{1}{2} & \frac{2}{3} \\
+ \frac{7}{6} & \frac{20}{3} & \frac{199}{36} \\
+ 1 & -\frac{71}{12} & \fbox{$-\frac{47}{18}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1831.txt

@@ -0,0 +1,222 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ \frac{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ \frac{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\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 }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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{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}
+ \frac{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{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: }\frac{2\ 13}{9\ 9}+\left(-\frac{10}{9}\right)\, \left(-\frac{8}{3}\right)=\frac{266}{81}. \\
+ \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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$\frac{266}{81}$} \\
+ \_ \\
+ \_ \\
+\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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{266}{81} \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 13}{3\ 9}+\left(-\frac{7}{9}\right)\, \left(-\frac{8}{3}\right)=\frac{82}{27}. \\
+ \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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{266}{81} \\
+ \fbox{$\frac{82}{27}$} \\
+ \_ \\
+\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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{266}{81} \\
+ \frac{82}{27} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{8}{9}\right)\, \times \, \frac{13}{9}+\left(-\frac{8}{9}\right)\, \left(-\frac{8}{3}\right)=\frac{88}{81}. \\
+ \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{2}{9} & -\frac{10}{9} \\
+ \frac{2}{3} & -\frac{7}{9} \\
+ -\frac{8}{9} & -\frac{8}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ \frac{13}{9} \\
+ -\frac{8}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{266}{81} \\
+ \frac{82}{27} \\
+ \fbox{$\frac{88}{81}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1920.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 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 }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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -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\ 2+(-1)\, \times \, 1=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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -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}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -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: }1\ 2+2\ 1+(-1)\, (-3)=7. \\
+ \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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & \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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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)\, \times \, 2+3\ 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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ \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}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, \times \, 2+(-1)\, \times \, 1+3 (-3)=-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}
+ \\
+ \left(
+\begin{array}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & \fbox{$-16$} \\
+ \_ & \_ \\
+\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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & -16 \\
+ \_ & \_ \\
+\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\ 1=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}
+ \\
+ \left(
+\begin{array}{ccc}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & -16 \\
+ \fbox{$4$} & \_ \\
+\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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & -16 \\
+ 4 & \_ \\
+\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+0 (-3)=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}
+ 1 & 2 & -1 \\
+ -3 & -1 & 3 \\
+ 2 & 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 2 & 1 \\
+ 1 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & 7 \\
+ -2 & -16 \\
+ 4 & \fbox{$5$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/1989.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -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 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}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 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}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 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: }(-3)\, (-3)+0\ 3=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}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$9$} \\
+ \_ \\
+\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 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 9 \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-3)+0\ 3=-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 1 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 9 \\
+ \fbox{$-3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2249.txt

@@ -0,0 +1,166 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 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 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 }2\times 1: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{ccc}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 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}{ccc}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 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: }(-2)\, (-3)+(-2)\, \times \, 3+3\ 3=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}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$9$} \\
+ \_ \\
+\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 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 9 \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-3)+2\ 3+(-1)\, \times \, 3=-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ -2 & -2 & 3 \\
+ 2 & 2 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -3 \\
+ 3 \\
+ 3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 9 \\
+ \fbox{$-3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2282.txt

@@ -0,0 +1,549 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{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{8}{3\ 3}+(-1)\, \times \, \frac{8}{3}+\frac{1}{3} (-0)=-\frac{16}{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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{16}{9}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & \_ & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{3}+(-1)\, \times \, 3+\frac{0 (-8)}{3}=-\frac{8}{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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & \fbox{$-\frac{8}{3}$} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & \_ \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }-\frac{7}{3\ 3}+(-1)\, \times \, 0+\frac{0\ 7}{3}=-\frac{7}{9}. \\
+ \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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & \fbox{$-\frac{7}{9}$} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \_ & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 8}{3\ 3}+\left(-\frac{2}{3}\right)\, \times \, \frac{8}{3}+\frac{2}{3}=\frac{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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \fbox{$\frac{2}{3}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{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{2}{3}+\left(-\frac{2}{3}\right)\, \times \, 3+(-2)\, \left(-\frac{8}{3}\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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & \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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & \_ \\
+ \_ & \_ & \_ \\
+\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{2}{3}\right)\, \times \, 0+(-2)\, \times \, \frac{7}{3}=-\frac{56}{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}
+ \\
+ \left(
+\begin{array}{ccc}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & \fbox{$-\frac{56}{9}$} \\
+ \_ & \_ & \_ \\
+\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}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4\ 8}{3\ 3}+\left(-\frac{2}{3}\right)\, \times \, \frac{8}{3}+\frac{1}{3} \left(-\frac{8}{3}\right)=\frac{8}{9}. \\
+ \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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \fbox{$\frac{8}{9}$} & \_ & \_ \\
+\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{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \frac{8}{9} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4}{3}+\left(-\frac{2}{3}\right)\, \times \, 3+\frac{8 (-8)}{3\ 3}=-\frac{70}{9}. \\
+ \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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \frac{8}{9} & \fbox{$-\frac{70}{9}$} & \_ \\
+\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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \frac{8}{9} & -\frac{70}{9} & \_ \\
+\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}+\left(-\frac{2}{3}\right)\, \times \, 0+\frac{8\ 7}{3\ 3}=\frac{28}{9}. \\
+ \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}
+ \frac{1}{3} & -1 & 0 \\
+ \frac{2}{3} & -\frac{2}{3} & -2 \\
+ \frac{4}{3} & -\frac{2}{3} & \frac{8}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ \frac{8}{3} & 1 & -\frac{7}{3} \\
+ \frac{8}{3} & 3 & 0 \\
+ -\frac{1}{3} & -\frac{8}{3} & \frac{7}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{16}{9} & -\frac{8}{3} & -\frac{7}{9} \\
+ \frac{2}{3} & 4 & -\frac{56}{9} \\
+ \frac{8}{9} & -\frac{70}{9} & \fbox{$\frac{28}{9}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/238.txt

@@ -0,0 +1,222 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right)$ 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}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\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 }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}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\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}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\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: }2 (-1)+(-1)\, (-3)=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 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ \_ \\
+ \_ \\
+\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)=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}{cc}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ \fbox{$8$} \\
+ \_ \\
+\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}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ 8 \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3 (-1)+0 (-3)=-3. \\
+ \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}
+ 2 & -1 \\
+ -2 & -2 \\
+ 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -1 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 1 \\
+ 8 \\
+ \fbox{$-3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2581.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 1 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 1 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{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 }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 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\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 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\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\ 3+\left(-\frac{13}{7}\right)\, \times \, \frac{11}{7}=\frac{4}{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}
+ 1 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$\frac{4}{49}$} \\
+ \_ \\
+\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}
+ 1 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{4}{49} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{20\ 3}{7}+\left(-\frac{13}{7}\right)\, \times \, \frac{11}{7}=\frac{277}{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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ 1 & -\frac{13}{7} \\
+ \frac{20}{7} & -\frac{13}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ \frac{11}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \frac{4}{49} \\
+ \fbox{$\frac{277}{49}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2642.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 0 \\
+ -1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 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 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}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ -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}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ -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: }(-2)\, \times \, 0+(-2)\, (-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}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ -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}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ -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: }(-2)\, \times \, 0+(-2)\, (-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ -2 & -2 \\
+ -2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 2 \\
+ \fbox{$2$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2649.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{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 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{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\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{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\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{5 (-4)}{7\ 7}+\frac{2}{7}+\frac{15\ 2}{7}=\frac{204}{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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$\frac{204}{49}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5\ 5}{7\ 7}+\frac{8}{7}+\frac{15 (-2)}{7\ 7}=\frac{51}{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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \fbox{$\frac{51}{49}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{16 (-4)}{7\ 7}+\left(-\frac{5}{7}\right)\, \times \, \frac{2}{7}+\left(-\frac{6}{7}\right)\, \times \, 2=-\frac{158}{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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ \fbox{$-\frac{158}{49}$} & \_ \\
+ \_ & \_ \\
+\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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{16\ 5}{7\ 7}+\left(-\frac{5}{7}\right)\, \times \, \frac{8}{7}+\left(-\frac{6}{7}\right)\, \left(-\frac{2}{7}\right)=\frac{52}{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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \fbox{$\frac{52}{49}$} \\
+ \_ & \_ \\
+\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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \frac{52}{49} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{20 (-4)}{7\ 7}+\left(-\frac{5}{7}\right)\, \times \, \frac{2}{7}+\frac{1}{7} (-2)=-\frac{104}{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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \frac{52}{49} \\
+ \fbox{$-\frac{104}{49}$} & \_ \\
+\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}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \frac{52}{49} \\
+ -\frac{104}{49} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{20\ 5}{7\ 7}+\left(-\frac{5}{7}\right)\, \times \, \frac{8}{7}+\frac{2}{7\ 7}=\frac{62}{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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ \frac{5}{7} & 1 & \frac{15}{7} \\
+ \frac{16}{7} & -\frac{5}{7} & -\frac{6}{7} \\
+ \frac{20}{7} & -\frac{5}{7} & -\frac{1}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{4}{7} & \frac{5}{7} \\
+ \frac{2}{7} & \frac{8}{7} \\
+ 2 & -\frac{2}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{204}{49} & \frac{51}{49} \\
+ -\frac{158}{49} & \frac{52}{49} \\
+ -\frac{104}{49} & \fbox{$\frac{62}{49}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2671.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -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}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -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}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -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)\, \times \, 0+0\ 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}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\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}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\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: }(-3)\, \times \, 0+0 (-2)=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}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & \fbox{$0$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-3)\, (-2)+0 (-3)=6. \\
+ \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 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & \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}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 6 \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 0+0\ 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}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 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}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 6 \\
+ 0 & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2\ 0+0 (-2)=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}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 6 \\
+ 0 & \fbox{$0$} & \_ \\
+\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 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 6 \\
+ 0 & 0 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+0 (-3)=-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 0 & 0 & -2 \\
+ 2 & -2 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ 0 & 0 & 6 \\
+ 0 & 0 & \fbox{$-4$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2742.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ -2 \\
+ -3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -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 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}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -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}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -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: }0 (-2)+0 (-3)=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}{cc}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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}{cc}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 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 (-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ 0 & 0 \\
+ -2 & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ -2 \\
+ -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 0 \\
+ \fbox{$1$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2757.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 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 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 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 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: }3 (-1)+2 (-1)=-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 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\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 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\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 (-3)+2\ 1=-7. \\
+ \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 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & \fbox{$-7$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }3\ 1+2\ 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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & \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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-1)+0 (-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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ \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}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ 2 & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-3)+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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ 2 & \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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ 2 & 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+0\ 2=-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}{cc}
+ 3 & 2 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -3 & 1 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -7 & 7 \\
+ 2 & 6 & \fbox{$-2$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2774.txt

@@ -0,0 +1,222 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 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 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}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -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}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -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: }1\ 3+(-2)\, (-2)=7. \\
+ \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 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 7 \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 3+0 (-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}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 7 \\
+ \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}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 7 \\
+ 0 \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 3+0 (-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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ 1 & -2 \\
+ 0 & 0 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 3 \\
+ -2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 7 \\
+ 0 \\
+ \fbox{$-6$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2779.txt

@@ -0,0 +1,166 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{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 }2\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 }2\times 1: \\
+\end{array}
+ \\
+ \left(
+\begin{array}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\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{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\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{2}{7}\right)\, \times \, 2+\left(-\frac{4}{7}\right)\, \times \, 1+\frac{17 (-8)}{7\ 7}=-\frac{192}{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}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \fbox{$-\frac{192}{49}$} \\
+ \_ \\
+\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}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{192}{49} \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{4\ 2}{7}+\frac{8}{7}+\frac{19 (-8)}{7\ 7}=-\frac{40}{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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{ccc}
+ -\frac{2}{7} & -\frac{4}{7} & \frac{17}{7} \\
+ \frac{4}{7} & \frac{8}{7} & \frac{19}{7} \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 2 \\
+ 1 \\
+ -\frac{8}{7} \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ -\frac{192}{49} \\
+ \fbox{$-\frac{40}{49}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2807.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \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 }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 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{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}{cc}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{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-\frac{7}{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}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \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}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \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{2\ 11}{4}+\frac{1}{4}=\frac{23}{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 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \fbox{$\frac{23}{4}$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }2 (-2)+\frac{1}{4\ 2}=-\frac{31}{8}. \\
+ \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 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & \fbox{$-\frac{31}{8}$} \\
+ \_ & \_ & \_ \\
+\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}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{4} (-2)+\frac{3 (-7)}{4}=-\frac{23}{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}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ \fbox{$-\frac{23}{4}$} & \_ & \_ \\
+\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}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ -\frac{23}{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{11}{4}\right)+3\ 1=\frac{37}{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}
+ \\
+ \left(
+\begin{array}{cc}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ -\frac{23}{4} & \fbox{$\frac{37}{16}$} & \_ \\
+\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 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ -\frac{23}{4} & \frac{37}{16} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2}{4}+\frac{3}{2}=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}{cc}
+ 2 & \frac{1}{4} \\
+ -\frac{1}{4} & 3 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 2 & \frac{11}{4} & -2 \\
+ -\frac{7}{4} & 1 & \frac{1}{2} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{57}{16} & \frac{23}{4} & -\frac{31}{8} \\
+ -\frac{23}{4} & \frac{37}{16} & \fbox{$2$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2883.txt

@@ -0,0 +1,159 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 0 \\
+ 2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 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 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}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 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}{cc}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 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: }0\ 0+1\ 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 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 2 \\
+\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}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 2 \\
+\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: }(-2)\, \times \, 0+0\ 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}
+ \\
+ \fbox{$
+\begin{array}{ll}
+ \text{Answer:} &  \\
+ \text{} & \left(
+\begin{array}{cc}
+ 0 & 1 \\
+ -2 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 2 \\
+ \fbox{$0$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/291.txt

@@ -0,0 +1,253 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -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 }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 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 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}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 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\ 0+0 (-1)=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}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\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}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\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 (-1)+0\ 0=-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 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 0 & \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 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 0 & -2 \\
+ \_ & \_ \\
+\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)\, (-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 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 0 & -2 \\
+ \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 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 0 & -2 \\
+ 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)\, \times \, 0=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}{cc}
+ 2 & 0 \\
+ -1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 0 & -1 \\
+ -1 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 0 & -2 \\
+ 1 & \fbox{$1$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/298.txt

@@ -0,0 +1,253 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)$.
+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}{cc}
+ -2 & -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}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -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}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -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: }1 (-2)+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}{cc}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -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}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -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: }1 (-2)+2 (-3)=-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}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & \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}{cc}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & -8 \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-2)+(-2)\, \times \, 3=-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}{cc}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & -8 \\
+ \fbox{$-8$} & \_ \\
+\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}{cc}
+ 1 & 2 \\
+ 1 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & -8 \\
+ -8 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1 (-2)+(-2)\, (-3)=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 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -2 & -2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 4 & -8 \\
+ -8 & \fbox{$4$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/2984.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -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 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 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -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 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -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)\, \times \, 3+1 (-1)+(-3)\, (-2)=-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}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \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}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-3)+1 (-1)+(-3)\, \times \, 0=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}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & \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}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0\ 3+3 (-1)+2 (-2)=-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}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ \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}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0 (-3)+3 (-1)+2\ 0=-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}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & \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}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & -3 \\
+ \_ & \_ \\
+\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)+(-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}
+ \\
+ \left(
+\begin{array}{ccc}
+ -2 & 1 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & -3 \\
+ \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 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & -3 \\
+ -2 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-1)\, (-3)+1 (-1)+(-1)\, \times \, 0=2. \\
+ \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 & -3 \\
+ 0 & 3 & 2 \\
+ -1 & 1 & -1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ 3 & -3 \\
+ -1 & -1 \\
+ -2 & 0 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -1 & 5 \\
+ -7 & -3 \\
+ -2 & \fbox{$2$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/302.txt

@@ -0,0 +1,231 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 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}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 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}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 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: }(-2)\, \times \, 0+0\ 0+2\ 2=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 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ \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}{ccc}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 4 \\
+ \_ \\
+ \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 0+2\ 0+(-2)\, \times \, 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 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 4 \\
+ \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 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 4 \\
+ -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+2\ 2=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}
+ -2 & 0 & 2 \\
+ 1 & 2 & -2 \\
+ -1 & 1 & 2 \\
+\end{array}
+\right).\left(
+\begin{array}{c}
+ 0 \\
+ 0 \\
+ 2 \\
+\end{array}
+\right)=\left(
+\begin{array}{c}
+ 4 \\
+ -4 \\
+ \fbox{$4$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3131.txt

@@ -0,0 +1,264 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -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 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}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -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)+0 (-1)+0\ 3=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}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\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)\, \times \, 3+0\ 2+0 (-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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\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 (-1)+2 (-1)+(-2)\, \times \, 3=-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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 2 & -6 \\
+ \fbox{$-9$} & \_ \\
+\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}{ccc}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 2 & -6 \\
+ -9 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }1\ 3+2\ 2+(-2)\, (-3)=13. \\
+ \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}
+ -2 & 0 & 0 \\
+ 1 & 2 & -2 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -1 & 3 \\
+ -1 & 2 \\
+ 3 & -3 \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ 2 & -6 \\
+ -9 & \fbox{$13$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3214.txt

@@ -0,0 +1,362 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{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 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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{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}{ccc}
+ -\frac{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{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: }\left(-\frac{35}{16}\right)\, \left(-\frac{21}{16}\right)+\left(-\frac{21}{16}\right)\, \left(-\frac{7}{16}\right)+\frac{33\ 33}{16\ 16}=\frac{1971}{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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$\frac{1971}{256}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{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)\, \left(-\frac{9}{4}\right)+\left(-\frac{21}{16}\right)\, (-3)+\frac{33}{16\ 8}=\frac{1167}{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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \fbox{$\frac{1167}{128}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \_ \\
+ \_ & \_ & \_ \\
+\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)\, \left(-\frac{43}{16}\right)+\left(-\frac{21}{16}\right)\, \times \, \frac{37}{16}+\frac{33 (-17)}{16\ 16}=\frac{167}{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}{ccc}
+ -\frac{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \fbox{$\frac{167}{256}$} \\
+ \_ & \_ & \_ \\
+\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}{ccc}
+ -\frac{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-21)}{2\ 16}+\left(-\frac{7}{8}\right)\, \left(-\frac{7}{16}\right)+\left(-\frac{23}{16}\right)\, \times \, \frac{33}{16}=-\frac{1165}{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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ \fbox{$-\frac{1165}{256}$} & \_ & \_ \\
+\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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ -\frac{1165}{256} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-9)}{2\ 4}+\left(-\frac{7}{8}\right)\, (-3)+\left(-\frac{23}{16}\right)\, \times \, \frac{1}{8}=-\frac{119}{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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ -\frac{1165}{256} & \fbox{$-\frac{119}{128}$} & \_ \\
+\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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ -\frac{1165}{256} & -\frac{119}{128} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-43)}{2\ 16}+\left(-\frac{7}{8}\right)\, \times \, \frac{37}{16}+\left(-\frac{23}{16}\right)\, \left(-\frac{17}{16}\right)=-\frac{1159}{256}. \\
+ \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{35}{16} & -\frac{21}{16} & \frac{33}{16} \\
+ \frac{3}{2} & -\frac{7}{8} & -\frac{23}{16} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{21}{16} & -\frac{9}{4} & -\frac{43}{16} \\
+ -\frac{7}{16} & -3 & \frac{37}{16} \\
+ \frac{33}{16} & \frac{1}{8} & -\frac{17}{16} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \frac{1971}{256} & \frac{1167}{128} & \frac{167}{256} \\
+ -\frac{1165}{256} & -\frac{119}{128} & \fbox{$-\frac{1159}{256}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3320.txt

@@ -0,0 +1,362 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{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 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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{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}{ccc}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{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: }0 (-1)+\left(-\frac{5}{2}\right)\, \times \, 2+0 (-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 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{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}{ccc}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{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: }\frac{0 (-3)}{2}+\left(-\frac{5}{2}\right)\, \times \, \frac{5}{2}+\frac{0 (-3)}{2}=-\frac{25}{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}{ccc}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & \fbox{$-\frac{25}{4}$} & \_ \\
+ \_ & \_ & \_ \\
+\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 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }0 (-1)+\frac{5}{2\ 2}+0 (-1)=\frac{5}{4}. \\
+ \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 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \fbox{$\frac{5}{4}$} \\
+ \_ & \_ & \_ \\
+\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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-1)}{2}+3\ 2+0 (-2)=\frac{9}{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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \fbox{$\frac{9}{2}$} & \_ & \_ \\
+\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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \frac{9}{2} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-3)}{2\ 2}+\frac{3\ 5}{2}+\frac{0 (-3)}{2}=\frac{21}{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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \frac{9}{2} & \fbox{$\frac{21}{4}$} & \_ \\
+\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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \frac{9}{2} & \frac{21}{4} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-1)}{2}+\frac{1}{2} (-3)+0 (-1)=-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}
+ 0 & -\frac{5}{2} & 0 \\
+ \frac{3}{2} & 3 & 0 \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -1 & -\frac{3}{2} & -1 \\
+ 2 & \frac{5}{2} & -\frac{1}{2} \\
+ -2 & -\frac{3}{2} & -1 \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -5 & -\frac{25}{4} & \frac{5}{4} \\
+ \frac{9}{2} & \frac{21}{4} & \fbox{$-3$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3336.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ \frac{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ \frac{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\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{5\ 26}{9\ 9}+\left(-\frac{4}{3}\right)\, \times \, 2+\frac{2}{3\ 9}=-\frac{80}{81}. \\
+ \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}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$-\frac{80}{81}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{5 (-7)}{9\ 3}+\left(-\frac{4}{3}\right)\, \left(-\frac{7}{9}\right)+\left(-\frac{2}{3}\right)\, \times \, \frac{13}{9}=-\frac{11}{9}. \\
+ \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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & \fbox{$-\frac{11}{9}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{26}{3\ 9}+\left(-\frac{22}{9}\right)\, \times \, 2+\frac{1}{9} (-3)=-\frac{115}{27}. \\
+ \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}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ \fbox{$-\frac{115}{27}$} & \_ \\
+ \_ & \_ \\
+\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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }-\frac{7}{3\ 3}+\left(-\frac{22}{9}\right)\, \left(-\frac{7}{9}\right)+\frac{3\ 13}{9}=\frac{442}{81}. \\
+ \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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \fbox{$\frac{442}{81}$} \\
+ \_ & \_ \\
+\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}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \frac{442}{81} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{3} \left(-\frac{26}{9}\right)+\frac{8\ 2}{3}+\frac{1}{9} \left(-\frac{20}{9}\right)=\frac{334}{81}. \\
+ \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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \frac{442}{81} \\
+ \fbox{$\frac{334}{81}$} & \_ \\
+\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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \frac{442}{81} \\
+ \frac{334}{81} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{7}{3\ 3}+\frac{8 (-7)}{3\ 9}+\frac{20\ 13}{9\ 9}=\frac{155}{81}. \\
+ \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{5}{9} & -\frac{4}{3} & -\frac{2}{3} \\
+ \frac{1}{3} & -\frac{22}{9} & 3 \\
+ -\frac{1}{3} & \frac{8}{3} & \frac{20}{9} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{26}{9} & -\frac{7}{3} \\
+ 2 & -\frac{7}{9} \\
+ -\frac{1}{9} & \frac{13}{9} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{80}{81} & -\frac{11}{9} \\
+ -\frac{115}{27} & \frac{442}{81} \\
+ \frac{334}{81} & \fbox{$\frac{155}{81}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3408.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ \frac{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ \frac{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\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{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\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{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\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{1}{10} \left(-\frac{3}{2}\right)+\left(-\frac{11}{10}\right)\, \times \, \frac{9}{5}=-\frac{213}{100}. \\
+ \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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{213}{100}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-2)}{2}+\left(-\frac{11}{10}\right)\, \times \, 3=-\frac{63}{10}. \\
+ \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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & \fbox{$-\frac{63}{10}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & \_ \\
+ \_ & \_ & \_ \\
+\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\ 5}+\left(-\frac{11}{10}\right)\, \times \, \frac{21}{10}=-\frac{561}{100}. \\
+ \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{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & \fbox{$-\frac{561}{100}$} \\
+ \_ & \_ & \_ \\
+\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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{10} \left(-\frac{19}{10}\right)+\left(-\frac{2}{5}\right)\, \times \, \frac{9}{5}=-\frac{91}{100}. \\
+ \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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ \fbox{$-\frac{91}{100}$} & \_ & \_ \\
+\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{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ -\frac{91}{100} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{19 (-2)}{10}+\left(-\frac{2}{5}\right)\, \times \, 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}
+ \\
+ \left(
+\begin{array}{cc}
+ \frac{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ -\frac{91}{100} & \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}{cc}
+ \frac{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ -\frac{91}{100} & -5 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{19 (-11)}{10\ 5}+\left(-\frac{2}{5}\right)\, \times \, \frac{21}{10}=-\frac{251}{50}. \\
+ \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{3}{2} & -\frac{11}{10} \\
+ \frac{19}{10} & -\frac{2}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ -\frac{1}{10} & -2 & -\frac{11}{5} \\
+ \frac{9}{5} & 3 & \frac{21}{10} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{213}{100} & -\frac{63}{10} & -\frac{561}{100} \\
+ -\frac{91}{100} & -5 & \fbox{$-\frac{251}{50}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3493.txt

@@ -0,0 +1,375 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{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 }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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{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}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{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{2}{5}+\frac{9 (-5)}{5\ 2}=-\frac{41}{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}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$-\frac{41}{10}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 3}{2}+\frac{9\ 9}{5\ 5}=\frac{156}{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}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \fbox{$\frac{156}{25}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}
+ 2 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{17}{10}\right)\, \times \, \frac{1}{5}+\left(-\frac{12}{5}\right)\, \left(-\frac{5}{2}\right)=\frac{283}{50}. \\
+ \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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \fbox{$\frac{283}{50}$} & \_ \\
+ \_ & \_ \\
+\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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{17}{10}\right)\, \times \, \frac{3}{2}+\left(-\frac{12}{5}\right)\, \times \, \frac{9}{5}=-\frac{687}{100}. \\
+ \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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & \fbox{$-\frac{687}{100}$} \\
+ \_ & \_ \\
+\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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & -\frac{687}{100} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{10}\right)\, \times \, \frac{1}{5}+\left(-\frac{9}{10}\right)\, \left(-\frac{5}{2}\right)=\frac{219}{100}. \\
+ \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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & -\frac{687}{100} \\
+ \fbox{$\frac{219}{100}$} & \_ \\
+\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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & -\frac{687}{100} \\
+ \frac{219}{100} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{3}{10}\right)\, \times \, \frac{3}{2}+\left(-\frac{9}{10}\right)\, \times \, \frac{9}{5}=-\frac{207}{100}. \\
+ \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 & \frac{9}{5} \\
+ -\frac{17}{10} & -\frac{12}{5} \\
+ -\frac{3}{10} & -\frac{9}{10} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{1}{5} & \frac{3}{2} \\
+ -\frac{5}{2} & \frac{9}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{41}{10} & \frac{156}{25} \\
+ \frac{283}{50} & -\frac{687}{100} \\
+ \frac{219}{100} & \fbox{$-\frac{207}{100}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3646.txt

@@ -0,0 +1,375 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ \frac{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ \frac{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{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 }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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{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{3 (-8)}{5\ 5}+\left(-\frac{6}{5}\right)\, \left(-\frac{14}{5}\right)=\frac{12}{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}
+ \frac{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$\frac{12}{5}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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}
+ \frac{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-11)}{5\ 5}+\left(-\frac{6}{5}\right)\, \times \, \frac{14}{5}=-\frac{117}{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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & \fbox{$-\frac{117}{25}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-8)}{5}+\left(-\frac{13}{5}\right)\, \left(-\frac{14}{5}\right)=\frac{62}{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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \fbox{$\frac{62}{25}$} & \_ \\
+ \_ & \_ \\
+\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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{3 (-11)}{5}+\left(-\frac{13}{5}\right)\, \times \, \frac{14}{5}=-\frac{347}{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}
+ \\
+ \left(
+\begin{array}{cc}
+ \frac{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & \fbox{$-\frac{347}{25}$} \\
+ \_ & \_ \\
+\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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & -\frac{347}{25} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{5}\right)\, \left(-\frac{8}{5}\right)+\frac{7 (-14)}{5\ 5}=-\frac{26}{25}. \\
+ \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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & -\frac{347}{25} \\
+ \fbox{$-\frac{26}{25}$} & \_ \\
+\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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & -\frac{347}{25} \\
+ -\frac{26}{25} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\left(-\frac{9}{5}\right)\, \left(-\frac{11}{5}\right)+\frac{7\ 14}{5\ 5}=\frac{197}{25}. \\
+ \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{3}{5} & -\frac{6}{5} \\
+ 3 & -\frac{13}{5} \\
+ -\frac{9}{5} & \frac{7}{5} \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ -\frac{8}{5} & -\frac{11}{5} \\
+ -\frac{14}{5} & \frac{14}{5} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \frac{12}{5} & -\frac{117}{25} \\
+ \frac{62}{25} & -\frac{347}{25} \\
+ -\frac{26}{25} & \fbox{$\frac{197}{25}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3680.txt

@@ -0,0 +1,390 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{ccc}
+ -\frac{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{ccc}
+ -\frac{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\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{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\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 \, \frac{4}{3}+\frac{5 (-11)}{6\ 6}-\frac{2}{3\ 3}=-\frac{119}{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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ \fbox{$-\frac{119}{36}$} & \_ \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{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)\, (-1)+\frac{5 (-5)}{6\ 6}+\frac{11}{3\ 6}=\frac{13}{12}. \\
+ \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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \fbox{$\frac{13}{12}$} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ \_ & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2\ 4}{3\ 3}+\left(-\frac{8}{3}\right)\, \left(-\frac{11}{6}\right)+\frac{2}{3\ 3}=6. \\
+ \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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ \fbox{$6$} & \_ \\
+ \_ & \_ \\
+\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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \_ \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2 (-1)}{3}+\left(-\frac{8}{3}\right)\, \left(-\frac{5}{6}\right)+\frac{1}{3} \left(-\frac{11}{6}\right)=\frac{17}{18}. \\
+ \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{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \fbox{$\frac{17}{18}$} \\
+ \_ & \_ \\
+\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{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \frac{17}{18} \\
+ \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{8\ 4}{3\ 3}+\left(-\frac{13}{6}\right)\, \left(-\frac{11}{6}\right)-\frac{2}{3}=\frac{247}{36}. \\
+ \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{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \frac{17}{18} \\
+ \fbox{$\frac{247}{36}$} & \_ \\
+\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{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \frac{17}{18} \\
+ \frac{247}{36} & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{8 (-1)}{3}+\left(-\frac{13}{6}\right)\, \left(-\frac{5}{6}\right)+\frac{11}{6}=\frac{35}{36}. \\
+ \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{7}{6} & \frac{5}{6} & \frac{1}{3} \\
+ \frac{2}{3} & -\frac{8}{3} & -\frac{1}{3} \\
+ \frac{8}{3} & -\frac{13}{6} & 1 \\
+\end{array}
+\right).\left(
+\begin{array}{cc}
+ \frac{4}{3} & -1 \\
+ -\frac{11}{6} & -\frac{5}{6} \\
+ -\frac{2}{3} & \frac{11}{6} \\
+\end{array}
+\right)=\left(
+\begin{array}{cc}
+ -\frac{119}{36} & \frac{13}{12} \\
+ 6 & \frac{17}{18} \\
+ \frac{247}{36} & \fbox{$\frac{35}{36}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}

pretraining/mathematica/linear_algebra/multiply_w_steps/3709.txt

@@ -0,0 +1,347 @@
+Problem:
+Multiply
+$\left(
+\begin{array}{cc}
+ -\frac{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right)$ and
+$\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)$.
+Answer:
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply the following matrices}: \\
+ \left(
+\begin{array}{cc}
+ -\frac{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{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 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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{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{1}{3} (-1)+\frac{1}{3} \left(-\frac{1}{3}\right)=-\frac{4}{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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ \fbox{$-\frac{4}{9}$} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & \_ & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{2}{3}-\frac{3}{3}=-\frac{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}
+ -\frac{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & \fbox{$-\frac{1}{3}$} & \_ \\
+ \_ & \_ & \_ \\
+\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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & \_ \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }\frac{1}{3} \left(-\frac{7}{3}\right)+\frac{1}{3\ 3}=-\frac{2}{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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & \fbox{$-\frac{2}{3}$} \\
+ \_ & \_ & \_ \\
+\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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ \_ & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, 1+\frac{5}{3\ 3}=-\frac{13}{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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ \fbox{$-\frac{13}{9}$} & \_ & \_ \\
+\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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ -\frac{13}{9} & \_ & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, (-2)+\left(-\frac{5}{3}\right)\, (-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}
+ \\
+ \left(
+\begin{array}{cc}
+ -\frac{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ -\frac{13}{9} & \fbox{$9$} & \_ \\
+\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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ -\frac{13}{9} & 9 & \_ \\
+\end{array}
+\right) \\
+\end{array}
+ \\
+ 
+\begin{array}{l}
+ 
+\begin{array}{l}
+ \text{Multiply }\text{corresponding }\text{components }\text{and }\text{add: }(-2)\, \times \, \frac{7}{3}+\left(-\frac{5}{3}\right)\, \times \, \frac{1}{3}=-\frac{47}{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{1}{3} & \frac{1}{3} \\
+ -2 & -\frac{5}{3} \\
+\end{array}
+\right).\left(
+\begin{array}{ccc}
+ 1 & -2 & \frac{7}{3} \\
+ -\frac{1}{3} & -3 & \frac{1}{3} \\
+\end{array}
+\right)=\left(
+\begin{array}{ccc}
+ -\frac{4}{9} & -\frac{1}{3} & -\frac{2}{3} \\
+ -\frac{13}{9} & 9 & \fbox{$-\frac{47}{9}$} \\
+\end{array}
+\right) \\
+\end{array}
+$} \\
+\end{array}
+ \\
+\end{array}