. Suppose x is an eigenvector of A corresponding to an eigenvalue 1. Show that x is an eigenvector of 5! A. What is the corresponding eigenvalue 1? (10 points) . Use mathematical induction to show that if A is an eigenvalue of an n x n matrix A, with x a corresponding eigenvector, then, for each positive integer m, 1'" is an eigenvalue of Am, with x a corresponding eigenvector. (15 points) . Show that if x is an eigenvector of the matrix product AB and Bx at (I, then Bx is an eigenvector of BA. (10 points) _ 0.4 0.3 . k 0.5 0.75 . LetAh 0.4 1.2] Explain whyA approaches[ 1-0 1.50]asl.:>m (15 points) . Suppose A = PEP1, where P is 2 x 2 and D = ['3 3] Let B = 51 3A +A2. Show that B is diagonalizable by nding a suitable factorization of B. (15 points) . If a, b, and c are distinct numbers, then the following system is inconsistent because the graphs of the equations are parallel planes. Show that the set of all least-squares solutions of the system is precisely the plane whose equation is \"*3\". (15 points) x2y+52=a x2y+52=b x2y+52=c x2y+52=