The paper proves upper bounds on the number of matrices with entries from a finite subset of a finite-rank multiplicative group that have a given rank, determinant, or characteristic polynomial.
Afifurrahman, ‘Some counting questions for matrix products ’, Bull
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Counting matrices over finite rank multiplicative groups
The paper proves upper bounds on the number of matrices with entries from a finite subset of a finite-rank multiplicative group that have a given rank, determinant, or characteristic polynomial.