# Counting matrices with different determinants

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!

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det A = det B is equivalent to the statement that log(A/B) is a traceless matrix; unlikely that this can be determined without an actual computation – Carlo Beenakker Sep 23 '11 at 13:21
Your counting problem reminds me a bit of the determinant spectrum problem indiana.edu/~maxdet/spectrum.html for which a little is known for $\pm1$ matrices. I think you want more even information though. – j.c. Sep 23 '11 at 17:34
I don't really understand the motivation: presumably the number of matrices with pairwise different determinants equals the number of possible determinants?! – Igor Rivin Sep 23 '11 at 19:57