$\det\left[ \begin{matrix} A & B \\ B & A \end{matrix} \right] \ = \ \det\left( A + B \right) \det( A - B )$ for $AB \neq BA$
How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved?