What I mean is that I don't know what is det(AB-BA) even if I do know det(AB) and det(BA).

I'm looking at

Sylvester's determinant theorem which looks related, but I still don't see a solution. Now I need to prove that for no M, det(M+I) = det(M)

~~, at least when M = AB..~~ (now that I think about it, there is probably no matrix that can't be written as a product of two others, is there?)