Let $A,B,C,D\in \mathbb{F}^{n×n}$ and let $M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$ have rank $n$. If $A$ is nonsingular, show that $M$ is equivalent to $\begin{bmatrix}A&B\\0&D-CA^{-1}B\end{bmatrix}$, and justify why $D-CA^{-1}B=0$.
Show that $det\left( \begin{bmatrix}|A|&|B|\\|C|&|D|\end{bmatrix}\right)=0$ using the above statement.
I have already shown the equivalence part. As well as what happens if $B,C,D$ are nonsingular. I haven't yet proven the $D-CA^{-1}B=0$ and I have to use all of these to prove that the determinant is zero.
Can you help me with this please? Thanks.