block matrices problem

block matrices problem

Let $A,B,C$ and $D$ be n by n matrics such that $AC=CA$. Prove that $\det
\begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(AD-CB)$.
The solution is to first assume that $A$ is invertible and then consider
the product
$$\begin{pmatrix} I & O\\ -CA^{-1} & I \end{pmatrix}\begin{pmatrix} A &
B\\ C & D \end{pmatrix}=\begin{pmatrix} A & B\\ O & D-CA^{-1}B
\end{pmatrix}$$
then it is not hard to prive that the claim is true if $A$ in invertible.
Finally, we use the fact that the set $GL_n$ form a dense open subset of
$M_n$ to get rid of the invertibility assumption. My question is: how to
come up with such a weired matrix $\begin{pmatrix} I & O\\ -CA^{-1} & I
\end{pmatrix}$? thank you so much. Is there any other problems that uses
the technique of assuming invertibility? (one of which i know is to prove
$det (I+AB)=det(I+BA)$), thanks in advance