1. The problem statement, all variables and given/known data Let A and B be diagonalizable 2 x 2 matrices. If every eigenvector of A is an eigenvector of B show that AB = BA. 2. Relevant equations D = PA(P^-1) 3. The attempt at a solution Since the eigenvectors are equivalent, wouldn't it hold true that P_A = P_B? If I have to show that AB = BA, I should be able to prove that PAB(P^-1) = PBA(P^-1) Since the eigenvectors of A are the eigenvectors of B, and P = (Eigenvector_1, Eigenvector_2) Then could I say that P_A = P_B, and (P^-1)_A = (P^-1)_B and then cancel out P and (P^-1) from the equation PAB(P^-1) = PBA(P^-1) and then conclude that AB=BA? Is my reasoning wrong here? Thanks a lot!