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Statistics of spectral properties of matrix-valued random variables.
49
votes
Expected determinant of a random NxN matrix
If $N \ge 2$, then the expected value is $0$ since interchanging two rows preserves the distribution but negates the determinant.
12
votes
Accepted
distribution of degree of minimum polynomial for eigenvalues of random matrix with elements ...
The survey article
Jason Fulman, Random matrix theory over finite fields, Bulletin of the AMS 39 (2002), 51-85
and the references therein should answer your questions to the extent that the answers …