A square matrix A is called nilpotent if Ak = O for some k ≥ 1.
(a) Prove that the only eigenvalue of a nilpotent matrix is 0. (The converse is also true; see Exercise 8.6.18.)
(b) Find examples where Ak-1 ≠ O but Ak = O when k = 2, 3, and in general.