Let $A,B\in M_n(\mathbb{R})$, Prove $\det\begin{pmatrix} A & -B\\ B & A \end{pmatrix}=|\det(A+iB)|^2$
Proving $\det \left( \begin{smallmatrix} A & -B \\ B & A \end{smallmatrix} \right) =|\det(A+iB)|^2$