Distribution of Matrices with Restricted Entries over Finite Fields

For a prime p, we consider some natural classes of matrices over a finite field Fp of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [−H,H] ⊆ [−(p − 1)/2,(p − 1)/2] is asymptotically close to the expected value.