Question 2 (6 marks) We wish to prove (without using Theorem 2.5) that if A, B, and A + B are n x n idempotent matrices, then AB = BA = 0. (a) Show that AB + BA = 0. (b) By Theorem 2.2, there exists a matrix P which diagonalises A: where r = =r(A). Write Ir 0 PT AP = D = 0 0 PT BP = A = A11 A12 A21 A22 where the block matrices are of the same dimensions as above. Show that A11 = 0, A12 = 0 and A21 = 0. (c) Show that AB = BA = 0.