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Given a set X of 
$n\times n$ matrices and a positive integer m, we consider the problem of estimating the cardinalities of the product sets 
$A_1 \cdots A_m$, where 
$A_i\in X$. When 
$X={\mathcal M}_n(\mathbb {Z};H)$, the set of 
$n\times n$ matrices with integer elements of size at most H, we give several bounds on the cardinalities of the product sets and solution sets of related equations such as 
$A_1 \cdots A_m=C$ and 
$A_1 \cdots A_m=B_1 \cdots B_m$. We also consider the case where X is the subset of matrices in 
${\mathcal M}_n(\mathbb {F})$, where 
$\mathbb {F}$ is a field with bounded rank 
$k\leq n$. In this case, we completely classify the related product set.
