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

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