1. The problem statement For integers m >= n, Prove det(xIm - AB) = xm-ndet(xIn - BA) for any x in R. 2. Relevant equations A is an m x n matrix B is an n x m matrix 3. The attempt at a solution I tried working out the characteristic polynomials by hand but it just seems too tedious for a nice proof. I know that each x is an eigenvalue of AB but after that I'm stumped.