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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.
$n-1$ Projection to the Nilpotent Operators on 
$\mathbb {C}^n$
Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in 
$\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank 
$n-1$ projection to the set of nilpotents in 
$\mathbb {M}_n(\mathbb {C})$ is 
$\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each 
$n\geq 2$, we construct examples of pairs 
$(Q,T)$ where Q is a projection of rank 
$n-1$ and 
$T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.
$2\times 2$ MATRICES
Let 
$T_{n}(\mathbb{F})$ be the semigroup of all upper triangular 
$n\times n$ matrices over a field 
$\mathbb{F}$. Let 
$UT_{n}(\mathbb{F})$ and 
$UT_{n}^{\pm 1}(\mathbb{F})$ be subsemigroups of 
$T_{n}(\mathbb{F})$, respectively, having 
$0$s and/or 
$1$s on the main diagonal and 
$0$s and/or 
$\pm 1$s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that 
$UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$ and 
$T_{2}(\mathbb{F})$ as involution semigroups under the skew transposition are nonfinitely based for any field 
$\mathbb{F}$.
