  We consider the set $\mathrm{M}_n(\mathbb{Z}; H))$ of $n\times n$-matrices
with integer elements of size at most $H$ and obtain upper and lower bounds on
the number of $s$-tuples of matrices from $\mathrm{M}_n(\mathbb{Z}; H)$,
satisfying various multiplicative relations, including multiplicative
dependence 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 obvious
distinction 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 the
asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to
be fixed and irreducible.
