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.

The Yang-Baxter-like matrix equation AXA = XAX is reconsidered, where A is any complex square matrix. A collection of spectral solutions for the unknown square matrix X were previously found. When A is diagonalisable, by applying the mean ergodic theorem we propose numerical methods to calculate those solutions.

The Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

If A is a nonsingular matrix such that its inverse is a stochastic matrix, the classic Brouwer fixed point theorem implies that the matrix equation AXA = XAX has a nontrivial solution. An explicit expression of this nontrivial solution is found via the mean ergodic theorem, and fixed point iteration is considered to find a nontrivial solution.

In this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).