Let A be a real n by n matrix satisfying A*A = I dn

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Let A be a real n by n matrix satisfying A*A = I dn. (a) Show, using only the second hypothesis, that the entries of any power A" of A are all bounded by 1 in absolute value. (b) Using this result7 show that the eigenvalues of A all have absolute value equal to 1. (0) Hence show that the only real eigenvalues that can occur at l and —1. (d) Suppose now det(A) = 1. Show that the product of the real eigenvalues is 1, if there are any. (e) Finally suppose n is odd and as well as det(A) = 1. Prove that 1 or —1 actually occurs as an eigenvalue.