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The 2 by 2 matrix [ [2, 0], [1, 2] ] has determinant 4, and just a single eigenvalue, 2.
Of course, if you want to raise the eigenvalues to the power of their algebraic multiplicities, then you do get the determinant. That's another nice way of thinking about it. But it wasn't what my tweet was referring too (geometric multiplicities).
Well we don't disagree then except on semantics. I think of n x n matrix always having n eigenvalues and Det is their product. I think that's very standard. But you're using a different convention.
The matrix I described above is 2 by 2, but has just a single eigenvector. So you're wrong about n x n matrices having n eigenvalues.
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