In order for the product XY to be defined, X has to have the same number of columns as Y has rows.

- AB is not defined since A has 2 columns but B has 3 rows.
- BA is defined since B has 3 columns and A has 3 rows.
- AC is not defined since C is the same size as B
- CA is defined since C has the same size as B.
- BC and CB are defined since B,C have the same size.

(I’ll go through 2 of these products since the other two are effectively the same with different numbers) We have:

\begin{align*}BA &= \begin{pmatrix}4& -3 & 1 \\ 3 & -2 & 1 \\ 7 & 0 & 1\end{pmatrix} \begin{pmatrix}-1 & 3 \\ 1 & 2 \\ 7 & -2\end{pmatrix} \\ & = \begin{pmatrix}-4 - 3 + 7 & 4 \times 3 - 6 - 2 \\ -3 - 2 +7 & 3 \times 3 - 4 - 2\\ -7 + 7 & 7 \times 3 - 2\end{pmatrix} \\ & = \begin{pmatrix}0 & 4 \\ 2 & 3 \\0 & 19\end{pmatrix} \end{align*}

Similarly:

CA = \begin{pmatrix}17 & -4 \\ 0 & 0 \\ 12 & 1\end{pmatrix}

We have:

\begin{align*}BC &= \begin{pmatrix}4&-3&1\\3&-2&1\\7&0&1\end{pmatrix} \begin{pmatrix}2&-2&3\\0&0&0\\5&-4&3\end{pmatrix} \\ & = \begin{pmatrix}4 \times 2 + 5 & -8 -4 & 12 + 3 \\ 3 \times 2 + 5 & -6 - 4 & 9 +3\\7 \times 2 + 5 & -14 - 4 & 21 + 3\end{pmatrix} \\ & = \begin{pmatrix}13 & -12 & 15 \\ 11 & -10 & 12 \\ 19 & -18 & 24\end{pmatrix}\end{align*}

Similarly:

CB = \begin{pmatrix}23 & -2 & 3 \\ 0 & 0 & 0 \\ 29 & -7 & 4\end{pmatrix}