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?
2026-03-25 19:05:18.1774465518
How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved?
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$$ \left( \begin{array}{cc} I&I \\ 0&I \\ \end{array} \right) \left( \begin{array}{cc} A&B \\ B&A \\ \end{array} \right) \left( \begin{array}{cc} I& -I \\ 0 &I \\ \end{array} \right) = \left( \begin{array}{cc} A+B& 0 \\ B& A-B \\ \end{array} \right) $$