3. Recall that for k E N, A is AA . . . A (k times). An n x n matrix is called nilpotent if Ak = O for some k E N. (a) (1 mark) Show that B = satisfies B2 = O. Thus B is nilpotent. This shows that there are nonzero nilpotent matrices. (b) (3 marks) Prove that every nilpotent matrix is singular. (c) (1 mark) Suppose A2 = O. Prove that I, + A is invertible with inverse In - A.. (d) (1 mark) Suppose A3 = O. Prove that In + A is invertible with inverse In - A + A2. (e) (3 marks) Suppose A is nilpotent. Prove that In + A is invertible and find the inverse of In + A. Prove that your inverse is correct