Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ divides $d_j(A)$, and this can easily be seen using determinental identities: given a $j\times j$ submatrix of $A$, perform successive Laplace expansions to express its determinant as a linear combination of $i\times i$ minors. Let $\alpha_i(A)=d_i(A)/d_{i-1}(A)$ be the invariant factors of $A$. From the existence of the Smith normal form of $A$, one has that $\alpha_i(A)$ divides $\alpha_j(A)$ whenever $i\leq j$, and it follows from this that $d_i(A)d_{j-1}(A)$ divides $d_{i-1}(A)d_j(A)$. I am interested in ways to show this latter result that do not appeal to the Smith normal form of $A$. Specifically:
Can one use determinental identities to show directly that $d_i(A)d_{j-1}(A)$ divides $d_{i-1}(A)d_j(A)$ whenever $i\leq j$?
For example, if one could express the product of an arbitrary $(i-1)\times(i-1)$ minor and an arbitrary $j\times j$ minor as a linear combination of products of $i\times i$ minors and $(j-1)\times(j-1)$ minors, the result would follow. Can this be done?
