Let $A$ be an $(m \times n)$ integer matrix (if it helps, we can assume that a is a square matrix). Let $d_i,\ldots,d_s$ be the elementary divisors of $A$. I am interested in the product $\prod_{i=1}^s d_i$. I know that I can determine this by looking at the gcd of the determinants of $(s \times s)$-submatrices. Is there any other way to determine this product? In particular I am interested in the case when $A$ is a square matrix with $\det(A)=0$ and I have no useful information about the eigenvalus of $A$.
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