Insight 1.4.24. Equal product of two matrices.
\begin{equation*}
\boldsymbol{A}\,\boldsymbol{B} = \boldsymbol{A}\,\boldsymbol{C} \text{ does not imply } \boldsymbol{B} = \boldsymbol{C}.
\end{equation*}
Example 1.4.25. Matrix product and equality.
Consider the following three matrices,
\begin{equation*}
\boldsymbol{A} = \left[ \begin{array}{ccc}
1 \amp -1 \amp 1\\
1 \amp 0 \amp 1\\
2 \amp 1 \amp 2\\
\end{array} \right], \hspace{1cm}
\boldsymbol{B} = \left[ \begin{array}{ccc}
1 \amp 2 \amp -1\\
0 \amp 1 \amp 2\\
1 \amp 1 \amp 1\\
\end{array} \right], \hspace{1cm}
\boldsymbol{C} = \left[ \begin{array}{ccc}
-1 \amp 3 \amp 1\\
0 \amp 1 \amp 2\\
3 \amp 0 \amp -1\\
\end{array} \right].
\end{equation*}
\begin{equation*}
\boldsymbol{A} \cdot \boldsymbol{B} =
\left[ \begin{array}{ccc}
2 \amp 2 \amp -2\\
2 \amp 3 \amp 0\\
4 \amp 7 \amp 2\\
\end{array} \right],
\end{equation*}
and
\begin{equation*}
\boldsymbol{A} \cdot \boldsymbol{C} =
\left[ \begin{array}{ccc}
2 \amp 2 \amp -2\\
2 \amp 3 \amp 0\\
4 \amp 7 \amp 2\\
\end{array} \right].
\end{equation*}
However, \(\boldsymbol{B} \ne \boldsymbol{C}. \)
