# Proof of the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$

by glS   Last Updated August 13, 2019 18:20 PM

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?

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

Will Jagy
August 13, 2019 18:15 PM

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