Answer
The matrix does not have an inverse.
Work Step by Step
Consider the given matrix $ A=\left[ \begin{matrix}
10 & -2 \\
-5 & 1 \\
\end{matrix} \right]$
Now, by using the inverse formula, we get:
${{A}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix}
d & -b \\
-c & a \\
\end{matrix} \right]$
Let, $\begin{align}
& a=10 \\
& b=-2 \\
& c=-5 \\
& d=1
\end{align}$
Substitute the values to get, $\begin{align}
& {{A}^{-1}}=\frac{1}{\left| ad-bc \right|}\left[ \begin{matrix}
d & -b \\
-c & a \\
\end{matrix} \right] \\
& {{A}^{-1}}=\frac{1}{\left| 10\times 1-\left( -2 \right)\times \left( -5 \right) \right|}\left[ \begin{matrix}
1 & 2 \\
5 & 10 \\
\end{matrix} \right] \\
& =\frac{1}{10-10}\left[ \begin{matrix}
2 & -3 \\
1 & 2 \\
\end{matrix} \right] \\
& =\frac{1}{0}\left[ \begin{matrix}
2 & -3 \\
1 & 2 \\
\end{matrix} \right]
\end{align}$
So, the matrix is not invertible.
Therefore, the matrix does not have an inverse because
$ ab-bc=0$
