Problem 8 (15 points). Let A, B E Rxxn a. (3 points) Prove that if I - AB is invertible then I - BA is invertible. Note: We do not know anything about invertibility of A and B. b. (3 points) Prove that (X, Y) = tr (X Y) defines an inner product, where X, Y E Rnxn and T denotes the transpose of a matrix. 3 c. (3 points) Suppose A E Rnxn is given. Prove that there is no X E Rnxn such that AX - XA = I. d. (2 point) Suppose A + B = AB. Show that AB = BA. e. (2 point) Suppose A, B E Rnxn are symmetric and skew-symmetric matrices respectively. Prove that tr(AB) = 0, where tr denotes the trace of a matrix. f. (2 point) Suppose A, B E RnAn are symmetric matrices. Prove that if Vx E R" : x] Ax = x Bx then A = B