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We consider the set $\mathrm{M}_n(\mathbb{Z}; H))$ of $n\times n$-matriceswith integer elements of size at most $H$ and obtain upper and lower bounds onthe number of $s$-tuples of matrices from $\mathrm{M}_n(\mathbb{Z}; H)$,satisfying various multiplicative relations, including multiplicativedependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb{Q})$.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 obviousdistinction due to the non-commutativity of matrices. As a part of our method,we obtain a new upper bound on the number of matrices from$\mathrm{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f\in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements theasymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has tobe fixed and irreducible.

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices