Answer
The product of $ AB={{I}_{4}}$, product of $ BA={{I}_{4}}$ and $ B $ is the multiplicative inverse of $ A $ $ B={{A}^{-1}}$.
Work Step by Step
The given expression is
$ A=\left[ \begin{matrix}
0 & 0 & -2 & 1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{matrix} \right],B=\left[ \begin{matrix}
1 & 2 & 0 & 3 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 2 & 0 & 2 \\
\end{matrix} \right]$
Now we will compute the matrix as $\left[ AB \right]$
$\begin{align}
& AB=\left[ \begin{matrix}
0 & 0 & -2 & 1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{matrix} \right]\left[ \begin{matrix}
1 & 2 & 0 & 3 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 2 & 0 & 2 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{matrix} \right] \\
& ={{I}_{4}}
\end{align}$ Andby computing the matrix as $\left[ BA \right]$ we get, $\begin{align}
& BA=\left[ \begin{matrix}
1 & 2 & 0 & 3 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 2 & 0 & 2 \\
\end{matrix} \right]\left[ \begin{matrix}
0 & 0 & -2 & 1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{matrix} \right] \\
& ={{I}_{4}}
\end{align}$
Both matrix $ AB $ and matrix $ BA $ are equal to the identity matrix
Now, we will compute the matrix $ B={{A}^{-1}}$.
$ B=\left[ \begin{matrix}
1 & 2 & 0 & 3 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 2 & 0 & 2 \\
\end{matrix} \right]$ And $ A=\left[ \begin{matrix}
0 & 0 & -2 & 1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{matrix} \right]$
Determine the inverse of matrix A.
Now, reduce matrix to row form:
$ A={{\left[ \begin{matrix}
0 & 0 & -2 & 1 \\
-1 & 0 & 1 & 1 \\
0 & 1 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{matrix} \right]}^{-1}}$
Augment with an identity matrix, $ A=\left[ \begin{matrix}
0 & 0 & -2 & 1 & 1 & 0 & 0 & 0 \\
-1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & -1 & 0 & 0 & 0 & 1 \\
\end{matrix} \right]$
Swap matrix rows:
$\begin{align}
& {{R}_{1}}\leftrightarrow {{R}_{2}} \\
& {{R}_{4}}\to {{R}_{4}}+1\times {{R}_{1}} \\
& {{R}_{2}}\leftrightarrow {{R}_{3}} \\
& {{R}_{4}}\to {{R}_{4}}+\frac{1}{2}\times {{R}_{3}}
\end{align}$
$ A=\left[ \begin{matrix}
-1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & -2 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} & 1 & 0 & 1 \\
\end{matrix} \right]$
Again
$\begin{align}
& {{R}_{4}}\to 2\times {{R}_{4}} \\
& {{R}_{3}}\to {{R}_{3}}-1\times {{R}_{4}} \\
& {{R}_{1}}\to {{R}_{1}}-1\times {{R}_{4}} \\
& {{R}_{3}}\to -\frac{1}{2}\times {{R}_{3}} \\
\end{align}$
$\left[ \begin{matrix}
-1 & 0 & 1 & 0 & -1 & -1 & 0 & -2 \\
0 & 1 & -1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 1 & 2 & 0 & 2 \\
\end{matrix} \right]$
Again:
$\begin{align}
& {{R}_{2}}\to {{R}_{2}}+1\times {{R}_{3}} \\
& {{R}_{1}}\to {{R}_{1}}-1\times {{R}_{3}} \\
& {{R}_{1}}\to -1\times {{R}_{1}} \\
\end{align}$
$\left[ \begin{matrix}
1 & 0 & 0 & 0 & 1 & 2 & 0 & 3 \\
0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 1 & 2 & 0 & 2 \\
\end{matrix} \right]$
Thus, ${{A}^{-1}}=\left[ \begin{matrix}
1 & 2 & 0 & 3 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 1 \\
1 & 2 & 0 & 2 \\
\end{matrix} \right]$
Therefore, matrix $ B $ is the multiplicative inverse of matrix $ A $