How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved

Question

The Wikipedia page about the determinant mentions the following matrix identity
$$\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B),$$
valid for squared matrices $A$ and $B$ of the same size.

How is this result proved?

Answer 1

$$ \left( \begin{array}{cc} I&I \\ 0&I \\ \end{array} \right) \left( \begin{array}{cc} A&B \\ B&A \\ \end{array} \right) \left( \begin{array}{cc} I& -I \\ 0 &I \\ \end{array} \right) = \left( \begin{array}{cc} A+B& 0 \\ B& A-B \\ \end{array} \right) $$