The task is to prove that for no two matrices A and B, A*B - B*A = I, where I is the identity matrix. I tried multiplying by the inverses of A or B, but that doesn't seem to lead to a more manageable form. The only way I see this could be done is by writing down all n*n (assuming n by n matrices) linear equations. It's easy to do when n = 2, but the same contradiction may not be as obvious for higher n. I hope there is a more intelligent way to go about this.
I know that det(AB) = det(BA), but I don't know what are the properties when subtraction is involved. Except for the case when only one line is different.
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)[STRIKE], at least when M = AB..[/STRIKE] (now that I think about it, there is probably no matrix that can't be written as a product of two others, is there?)