Decide whether the following statements are true or false. If a statement is true then give a proof. If a statement is false then give a counterexample. (a) For any A C"x" with eigenvalues A, A2,..., An, one has det (A) = II, 1j. (b) If A Cnx admits the decomposition A = UAU* with U unitary and A diagonal then A is normal (i.e., AA* = A*A). (c) Every normal matrix has distinct eigenvalues. (d) The eigenvalues of a skew-Hermitian matrix (A* = -A) are pure imaginary (i.e., their real parts are zero). (e) If x, y ER" are such that ||zy||2||x+y||2 then ry = 0.