Linear Algebra - Real Matrix and Invertibility Let $M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$ be a real matrix $2n\times 2n$ with $A,B,C,D$ real matrices $n\times n$ that are commutative to each other. Show that $M$ is invertible if and only if $AD-BC$ is invertible.

Hint: try considering
$$
\begin{pmatrix}
D(AD-BC)^{-1} & -B(AD-BC)^{-1} \\
-C(AD-BC)^{-1} & A(AD-BC)^{-1}
\end{pmatrix}
$$
