Consider the following. 2 1 U 0 1 3 A = 0 2 D r p = 0 4 0 4 2 4 1 2 2 (a) Verify that A is diagonalizable by computing P'lAP. l-1 l l0 l l0 l Vim): o -2 0 : Jlii X (b) Use the result of part (a) and the theorem below to find the eigenvalues ofA. Similar Matrices Have the Same Eigenvalues If}! and B are similar n x n matrices; then they have the same eigenvalues. WlS) For the matrix A, find (if possible) a nonsingular matrix P such that P'lAP is diagonal. (If not possible, enter IMPOSSIBLE.) 100 A: 421 102 | || || | .3 P=| || || ls, ggg w Verify that P'lAP is a diagonal matrix with the eigenvalues on the main diagonal. .3 P'lAP= | || || | , iil w Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = P-AP is the diagonal form of A. Prove that AK = PB*p-1, where k is a positive integer. Use the result above to find the indicated power of A. A = -2 -6 46 2 5 253/64 189/32 46 -63/32 -47/16