Answer
The inverse of the matrix $ A $ is, ${{A}^{-1}}=\left[ \begin{matrix}
-\frac{3}{5} & \frac{1}{5} \\
1 & 0 \\
\end{matrix} \right]$
Work Step by Step
At first we will find,
$\begin{align}
& \det A=0\left( 3 \right)-1\left( 5 \right) \\
& =0-5 \\
& =-5
\end{align}$
The inverse of the matrix $ A $ is calculated as below:
$\begin{align}
& {{A}^{-1}}=\frac{1}{\det A}\left[ \begin{matrix}
3 & -1 \\
-5 & 0 \\
\end{matrix} \right] \\
& =\frac{1}{-5}\left[ \begin{matrix}
3 & -1 \\
-5 & 0 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-\frac{3}{5} & \frac{1}{5} \\
1 & 0 \\
\end{matrix} \right]
\end{align}$
Now we will consider,
$\begin{align}
& A{{A}^{-1}}=\left[ \begin{matrix}
0 & 1 \\
5 & 3 \\
\end{matrix} \right]\left[ \begin{matrix}
-\frac{3}{5} & \frac{1}{5} \\
1 & 0 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
0\left( -\frac{3}{5} \right)+1 & 0\left( \frac{1}{5} \right)+1\left( 0 \right) \\
5\left( -\frac{3}{5} \right)+3\left( 1 \right) & 5\left( \frac{1}{5} \right)+3\left( 0 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]
\end{align}$
Next we will consider,
$\begin{align}
& {{A}^{-1}}A=\left[ \begin{matrix}
-\frac{3}{5} & \frac{1}{5} \\
1 & 0 \\
\end{matrix} \right]\left[ \begin{matrix}
0 & 1 \\
5 & 3 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
-\frac{3}{5}\left( 0 \right)+\frac{1}{5}\left( 5 \right) & -\frac{3}{5}\left( 1 \right)+\frac{1}{5}\left( 3 \right) \\
1\left( 0 \right)+0\left( 5 \right) & 1\left( 1 \right)+0\left( 3 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]
\end{align}$
Clearly, $ A{{A}^{-1}}={{A}^{-1}}A=I $
Hence, ${{A}^{-1}}=\left[ \begin{matrix}
-\frac{3}{5} & \frac{1}{5} \\
1 & 0 \\
\end{matrix} \right]$.
