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I was studying block matrices and suddenly this question came to my mind.

Let $A, B \in \Bbb R^{n \times n}$. From this Wikipedia page,

$$\det \begin{pmatrix} A & B\\ B & A\end{pmatrix} = \det(A-B)\det(A+B)$$

even if $A$ and $B$ do not commute. Does a similar condition hold for the following block matrix?

$$\begin{pmatrix} A & -B\\ B & A \end{pmatrix}$$