All 2×2 matrices satisfy the following equalitypic.twitter.com/6vlgtm8JXa
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In general, for an nxn matrix M, write det(M-λI) as an n-degree polynomial in λ then let λ=M and set it to 0
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exp(tr(X)) = det(exp(X)) IIRC is another great one that involves eigenvalues (whose sum is equal to the trace, and whose product is equal to the determinant)
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Yes I believe I tweeted that around a month ago
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