Answer
$A B=\left[\begin{array}{cc|c}3 & 2 & 0 \\ -1 & 1 & 0 \\ \hline 0 & 0 & 5\end{array}\right]$
Work Step by Step
Denote the blocks of the matrix A by $A_{1}, A_{2}, A_{3},$ and $A_{4}$,
\[
A=\left[\begin{array}{cc|cc}
1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\hline 0 & 0 & 2 & 3
\end{array}\right] \quad B=\left[\begin{array}{cc|c}
2 & 3 & 0 \\
-1 & 1 & 0 \\
\hline 0 & 0 & 1 \\
0 & 0 & 1
\end{array}\right]
\]
and the blocks of the matrix $\mathrm{B}$ by $B_{1}, B_{2}, B_{3},$ and $B_{4}$
\[
\begin{aligned}
A=\left[\begin{array}{cc}
A_{1} & A_{2} \\
A_{3} & A_{4}
\end{array}\right] B=\left[\begin{array}{cc}
B_{1} & B_{2} \\
B_{3} & B_{4}
\end{array}\right] \\
A B=\left[\begin{array}{cc}
A_{1} B_{1}+A_{2} B_{3} & A_{1} B_{2}+A_{2} B_{4} \\
A_{3} B_{1}+A_{4} B_{3} & A_{3} B_{2}+A_{4} B_{4}
\end{array}\right] & \text { Calculate } A_{1} B_{1}+A_{2} B_{3}
\end{aligned}
\]
$A_{1} B_{1}+A_{2} B_{3}=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}2 & 3 \\ -1 & 1\end{array}\right]+\left[\begin{array}{cc}0 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{cc}0 & 0 \\ 0 & 0\end{array}\right]=$ Calculate $A_{1} B_{2}+A_{2} B_{4}$
$\left[\begin{array}{cc}3 & 2 \\ -1 & 1\end{array}\right]$
Calculate $A_{1} B_{2}+A_{2} B_{4}=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}0 \\ 0\end{array}\right]+\left[\begin{array}{cc}0 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]$ $A_{3} B_{1}+A_{4} B_{3}$
Calculate $A_{3} B_{1}+A_{4} B_{3}=\left[\begin{array}{ll}0 & 0\end{array}\right]\left[\begin{array}{cc}2 & 3 \\ -1 & 1\end{array}\right]+\left[\begin{array}{cc}2 & 3\end{array}\right]\left[\begin{array}{cc}0 & 0 \\ 0 & 0\end{array}\right]=$ $A_{3} B_{2}+A_{4} B_{4}$
0
$A_{3} B_{1}+A_{4} B_{3}=\left[\begin{array}{ll}0 & 0\end{array}\right]\left[\begin{array}{l}0 \\ 0\end{array}\right]+\left[\begin{array}{ll}2 & 3\end{array}\right]\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{ll}5\end{array}\right] \quad \begin{array}{c}\text { Back substitute these values into the formula of the } \\ \text { product } \mathrm{AB}\end{array}$
