Answer
See answer below
Work Step by Step
Suppose that A and B are invertible matrices:
$\begin{pmatrix}
A& 0\\
0 & B^{-1}
\end{pmatrix}=\begin{pmatrix}
A^{-1}& 0\\
0 & B
\end{pmatrix}$
Hence $\begin{pmatrix}
A& 0\\
0 & B^{-1}
\end{pmatrix}\begin{pmatrix}
A^{-1}& 0\\
0 & B
\end{pmatrix}=\begin{pmatrix}
I_n& 0\\
0 & I_m
\end{pmatrix}=I_{n+m}$
and $\begin{pmatrix}
A^{-1}& 0\\
0 & B
\end{pmatrix}\begin{pmatrix}
A& 0\\
0 & B^{-1}
\end{pmatrix}=\begin{pmatrix}
I_n& 0\\
0 & I_m
\end{pmatrix}=I_{n+m}$
Therefore, the given block matrix is invertible.
