Let $A$ and $B$ be two matrices of order $n$ over a finite subset of integers $S$ such that $A$ and $B$ are positive-definite, nonsingular and symmetric.

I am interested in proprieties about $A$, $B$ that would allow me to conclude that $det(A) \not = det(B),$ without actually computing the determinant.

Motivation: I would like to bound (from bellow) the number of such matrices (having some additional structure) such that they have mutually different determinants.

I assume this problem is hard in general, but any pointers to relevant literature would be appreciated.

Thanks!