Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$
Solution 1:
Amazingly, this is exactly Lemma 3 in my paper "Almost all integer matrices have no integer eigenvalues" with Erick B. Wong (other than restricting to $i=1,j=2$ which is trivially generalized). There's a short proof there which seems to fit your needs. There's also a reference to a book on curious determinant identities of this sort in the bibliography (which is where we got the proof in the first place).