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.
2026-04-08 11:33:08.1775647988
Linear Algebra - Real Matrix and Invertibility
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Hint: try considering $$ \begin{pmatrix} D(AD-BC)^{-1} & -B(AD-BC)^{-1} \\ -C(AD-BC)^{-1} & A(AD-BC)^{-1} \end{pmatrix} $$