(i) Let A be an n × n symmetric matrix such that A and I_n – A are both positive semi-definite. Show that 0 ≤ a_ii ≤ 1 for i = 1, . . . , n, where α_ii is the i^th diagonal element of A.

(ii) Prove that if A is an n × n symmetric, idempotent matrix then it must be positive semi-definite.

(iii) Prove that the only n × n symmetric, idempotent matrix that is also invertible is I_n.