We consider the set Mn (Z; H )) of n × n-matrices with integer elements of size at most H and obtain a new upper bound on the number of matrices from Mn(Z; H) with a given characteristic polynomial f ∈ Z[X], which is uniform with respect to f. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which f has to be fixed and irreducible. We use our result to address various other questions of arithmetic statistics for matrices from Mn(Z;H), eg satisfying certain multiplicative relations. Some of these problems generalise those studied in the scalar case n = 1 by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices.

Joint works with Kamil Bulinski, Philipp Habegger and Igor Shparlinski.