(a) Let M and N be n n (n 2) square matrices such that M = (mij ) has entries mij = { a if i = j 0 if i = j and N = (nij ) has entries nij = 1 for all i, j.

(i) Show that N2 = nN.

(ii) Express M in terms of N and the identity matrix I.

(iii) Use (ii) to show that M is invertible when a = 0 and find M1 . (Hint: Find M1 in terms of N and I.)

(b) A square matrix is said to be nilpotent if there exists a positive integer m such that A^m = 0. Prove that if A is nilpotent and B is an invertible matrix of order n such that AB = BA, then (B A) is invertible. (Hint: Consider the expression B^mA^m and make use of Theorem 2.4.14.)