Let A and B be 2x2 matrices with complex entries such that AB-BA is a linear combination of A and B. Prove that A and B share a common nonzero eigenvector.
I manipulated the given so that A(B-yI)=B(A-xI), where x and y are scalars, but I'm unsure how to get from there to showing that there is a common eigenvector (the null of (B-yI) and (A-xI) are not disjoint).