# Number of matrix factorizations [closed]

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?)

• @QiaochuYuan Yes Aug 5 at 18:42
• 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. Aug 5 at 19:34
• @ThomasPreu I assume here that it is allowed that $S_i = S_j$ for distinct $i, j$. Aug 5 at 20:18