Answer
Matrix $ B $ is not a multiplicative inverse of the matrix $ A $.
Work Step by Step
Find the product of $ AB $ as follows:
$\begin{align}
& AB=\left[ \begin{matrix}
2 & 7 \\
1 & 4 \\
\end{matrix} \right]\left[ \begin{matrix}
4 & -7 \\
-1 & 3 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
2\left( 4 \right)+7\left( -1 \right) & 2\left( -7 \right)+7\left( 3 \right) \\
1\left( 4 \right)+4\left( -1 \right) & 1\left( -7 \right)+4\left( 3 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
8-7 & -14+21 \\
4-4 & -7+12 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 7 \\
0 & 5 \\
\end{matrix} \right]
\end{align}$
Therefore, $ AB=\left[ \begin{matrix}
1 & 7 \\
0 & 5 \\
\end{matrix} \right]$
Next we will find the product of $ BA $ as follows:
$\begin{align}
& BA=\left[ \begin{matrix}
4 & -7 \\
-1 & 3 \\
\end{matrix} \right]\left[ \begin{matrix}
2 & 7 \\
1 & 4 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
4\left( 2 \right)-7\left( 1 \right) & 4\left( 7 \right)-7\left( 4 \right) \\
-1\left( 2 \right)+3\left( 1 \right) & -1\left( 7 \right)+3\left( 4 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
8-7 & 28-28 \\
-2+3 & -7+12 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 \\
1 & 5 \\
\end{matrix} \right]
\end{align}$
As $ AB\ne BA\ne I $, where x is an identity matrix, therefore, the given matrix is not a multiplicative inverse of the matrix .
