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Prove the following statements: a) If A and B are two positive semidefinite matrices in IR...

Prove the following statements:

a) If A and B are two positive semidefinite matrices in IR ^ n × n , then trace (AB) ≥ 0. If, in addition, trace (AB) = 0, then AB = BA =0

b) Let A and B be two (different) n × n real matrices such that R(A) = R(B), where R(·) denotes the range of a matrix.
(1) Show that R(A + B) is a subspace of R(A).
(2) Is it always true that R(A + B) = R(A)? If so, prove it; otherwise, give a counterexample

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