(i) Let A be an n 3 n symmetric matrix such
that A and In 2 A are both positive semi-definite. Show that 0 # aii # 1 for i
5 1, . . . , n, where aii is the i th diagonal element of A. (ii) Prove that if
A is an n 3 n symmetric, idempotent matrix then it must be positive semi-definite.
(iii) Prove that the only n 3 n symmetric, idempotent matrix that is also
invertible is In.
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