Answer
See work
Work Step by Step
We will Multiply matrices on left side:
$\left[\begin{array}{cc}X & 0 \\ Y & Z\end{array}\right] \cdot\left[\begin{array}{cc}A & 0 \\ B & C\end{array}\right]=\left[\begin{array}{cc}X A & 0 \\ Y A+Z B & Z C\end{array}\right]$
Now this matrix equals matrix on the right side
$\left[\begin{array}{cc}X A & 0 \\ Y A+Z B & Z C\end{array}\right]=\left[\begin{array}{cc}I & 0 \\ 0 & I\end{array}\right]$
$X A=I \rightarrow X=I \cdot A^{-1}=A^{-1}$
$Z C=I \rightarrow Z=C^{-1} \cdot I=C^{-1}$
$Y A+Z B=0$
$Y A=-Z B$
$Y=-Z B A^{-1}=-C^{-1} B A^{-1}$
