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How to prove this determinant is positive?

Given matrices $$A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr)$$ where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following? $$\det \big( I + e^{A_1}e^ …
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