Answer
See the explanation below.
Work Step by Step
Consider the two matrices $A$ and $B$ be the multiplicative inverses; that is, the product of the matrix AB and $BA$ is equal to the identity matrix.
As example:
Let matrix $A=\left[ \begin{matrix}
4 & -3 \\
-5 & 4 \\
\end{matrix} \right]$ and $B=\left[ \begin{matrix}
4 & 3 \\
5 & 4 \\
\end{matrix} \right]$
If $B$ is the multiplicative inverse of $A$, then both $AB$ and $BA$ must result in the identity matrix
${{I}_{2}}=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]$
Checking whether $AB={{I}_{2}}$
$\begin{align}
& AB={{I}_{2}} \\
& AB=\left[ \begin{matrix}
4 & -3 \\
-5 & 4 \\
\end{matrix} \right]\left[ \begin{matrix}
4 & 3 \\
5 & 4 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \\
& ={{I}_{2}}
\end{align}$
Now, check whether $BA={{I}_{2}}$
$\begin{align}
& BA=\left[ \begin{matrix}
4 & 3 \\
5 & 4 \\
\end{matrix} \right]\left[ \begin{matrix}
4 & -3 \\
-5 & 4 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \\
& ={{I}_{2}}
\end{align}$
Thus, matrix $B$ is the multiplicative inverse of matrix $A$. Hence, $AB={{I}_{n}}$ and $BA={{I}_{n}}$.
