Let be an matrix over a field .

Let , and let be injective, monotone functions. (That is, .) The -minor of , denoted , is the matrix whose entry is the entry of . We will call such and *crunchy*.

Let be the unique monic generator in of the ideal .

Now suppose is the Smith normal form of , and that the diagonal entries of are . Prove that , taking .

(Setting is in fact the proper thing to do. Think of a matrix as a function from to ; then if , this function is empty. Now , and in the formula , the empty product is 1, and we have one summand. So the ideal in is .)

[I consulted these notes by Gregg Musiker for this problem.]

We begin by arguing that , where is an elementary matrix. (We discussed elementary matrices previously.) Recall that left multiplication by an elementary matrix corresponds to one of the three elementary row operations. Let and be crunchy, and let and .

Suppose interchanges rows and .

- If , then , and so .
- If , then is obtained from by interchanging rows and . So we have for some row-swapping elementary matrix , and thus .
- Suppose and . Now let be the (unique) crunchy function whose image is with replaced by . Now is obtained from by swapping some rows. So , where is some product of row-swapping elementary matrices, and we have for some .

Thus we have , and so .

Now suppose multiplies row by a field element .

- If , then , and so .
- If , then is obtained from by multiplying row by . So we have .

Thus we have , and so .

Finally, suppose adds times row to row .

- If , then , and so .
- If and , then is obtained from by adding a multiple of row to row . So for some row-adding elementary matrix , and we have .
- Suppose and . Now is obtained from by adding some unrelated row vector to row . In particular, we have , where is a product of two elementary matrices; one which swaps rows and , and one which multiplies the (new) th row by . We’ve already seen that is a unit multiple of .

Thus we have , and so .

So for any elementary matrix , . Now note that , so that . So we also have for elementary matrices .

By Theorem 21 in D&F, we have in Smith Normal Form, where and are products of elementary matrices. In particular, .

Now we claim that .

Claim: If , then . To see this, Note that if we remove some row from a diagonal matrix, then the result has a zero column. Unless we also remove the corresponding column, the determinant is 0.

Now suppose is crunchy on . By the divisibility condition on the diagonal entries of , we have for all . In particular, the determinant of the minor with for divides the determinant of every other minor. So , and follows.