Two $n \times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1} A S$ and $S^{-1} B S$ are diagonal matrices.
(a) Show that simultaneously diagonalizable matrices commute: $A B=B A$.
(b) Prove that the converse is valid, provided that one of the matrices has no multiple eigenvalues. (c) Is every pair of commuting matrices simultaneously diagonalizable?