[12 points] Consider the matrix ;1_3 30 48 TR L48 11 (a) [2 points] Show that if 1s an eigenvector with eigenvalue A of any matrix I3, then v 1s also an eigenvector of the matrix (B) for any constant [R. What is corresponding eigenvalue of 37? (b) [3 points] Hence or otherwise, state the characteristic polynomial and find the eigenvalues of A. (c) [3 points] Find the ser of all eigenvectors for A corresponding to each of the eigenvalues. (d) [2 points] Show that the eigenvectors for the two different eigenvalues are orthogonal. Then select suitable eigenvectors so that the matrix P = (v | w) (i.e. the matrix formed by two eigenvectors in columns of 1) T . . v . . . : X has inverse V! = V7T = wT (i.e. inverse formed by the same two eigenvectors in rows of V1) (e) [2 points] For any n x n matrix B, if B = PDP1 with all diagonal entries [);; of [J non-zero, show that B~' = PD~'P~! where D~ is a diagonal matrix with diagonal entries 1. Hence find A~ as the product of 3 matrices (no need to multply out the matrices)