Proposition 6

— Let X be a matrix of rank r over a principal ideal domain A, and let $A \alpha_i$ ($1 \leq i \leq r$) be its sequence of invariant factors. Then $\alpha_1$ is a gcd of the elements of X; and the product $\alpha_1 \ldots \alpha_q$ is a gcd of the q-th order minors of X for $q \leq r$.