Answer
See explanation
Work Step by Step
Let $A$ is invertible, and let $A^{-1}=\left[\begin{array}{ll}D & E \\ F & G\end{array}\right]$ Then $A A^{-1}=\left[\begin{array}{ll}B & 0 \\ 0 & C\end{array}\right] \cdot\left[\begin{array}{ll}D & E \\ F & G\end{array}\right]=\left[\begin{array}{ll}B D & B E \\ C F & C G\end{array}\right]=\left[\begin{array}{ll}I & 0 \\ 0 & I\end{array}\right]$
So,
$B D=I$ and $C G=I$, meaning that $B, C$ are invertible.
If $B, C$ are invertible, try $\left[\begin{array}{cc}B^{-1} & 0 \\ 0 & C^{-1}\end{array}\right]$ for inverse of $A$
$\left[\begin{array}{cc}B & 0 \\ 0 & C\end{array}\right] \cdot\left[\begin{array}{cc}B^{-1} & 0 \\ 0 & C^{-1}\end{array}\right]=\left[\begin{array}{cc}B B^{-1} & 0 \\ 0 & C C^{-1}\end{array}\right]=\left[\begin{array}{cc}I & 0 \\ 0 & I\end{array}\right]$
so $A$ is invertible.
