(i) Let A be an n x n symmetric matrix such that A and I, – A are both positive semi-definite. Show that 0 < a s 1 for i = 1, . .., n, where a is the ih 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 X n symmetric, idempotent matrix that is also invertible is I.

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