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Let $X$ be the set of $2N+1$ integers from $-N$ to $N$. Let $M = X^{k \times k}$ be the set of all $k \times k$ matrices with entries taken from $X$. My question is the following.

Let $A \in \mathbb{Z}^{k \times k}$. Then for fixed $n$, how many collections $\{S_1,...,S_n\}$ where $S_i \in M$, are there such that there is an order of multiplication of the elements in $\{S_1,...,S_n\}$ such that the product equals $A$? (If such a product exists at all?)

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  • $\begingroup$ @QiaochuYuan Yes $\endgroup$
    – user918212
    Aug 5 at 18:42
  • $\begingroup$ Is this "collection" a set or a multiset? I.e. may $S_i=S_j$ for $i\neq j$ or not? In any case the factorization of the determinant should play a role and also $\text{SL}_k(\mathbb{Z})$. Without the height bound of $N$ for the matrix entries the result would be $\infty$ for $k\geq 2$ and $n\geq 3$, because $|\text{SL}_k(\mathbb{Z})|=\infty$. The height bound makes this problem non-trivial. $\endgroup$ Aug 5 at 19:34
  • $\begingroup$ @ThomasPreu I assume here that it is allowed that $S_i = S_j$ for distinct $i, j$. $\endgroup$
    – user918212
    Aug 5 at 20:18

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