A square matrix A is said to be nilpotent of index k if there exist a smallest positive integer k such that Ak = 0. (a) What is the index of the nilpotent matrix {(0,1,2)t,(1,0,0 )t ,(2,1,0)t }?
(b) If A is nilpotent, what can you say about AT ? Prove your answer.
(c) Are nilpotent matrices invertible? Prove your answer.
(d) If A is nilpotent, is I − A invertible? Prove your answer.
