Let X and ff denote, respectively, the mean and the variance of a random sample of size n from a distribution which is N(O, u2).

(a) IfA denotes the symmetric matrix of n.X2, show that A = (1/n)P,where P is the n x n matrix, each of whose elements is equal to one.

(b) Demonstrate that A is idempotent and that the tr A = 1. Thus nX 2/tr is x2(1).

(c) Show that the symmetric matrix B of nff is I - (1/n)P.

(d) Demonstrate that B is idempotent and that tr B = n - l. Thus nS2/a2 is x2(n - 1), as previously proved otherwise.

(e) Show that the product matrix AB is the zero matrix.