Use the method of Example 4.29 to compute the indicated power of the matrix. [ " T Example 4.29 Compute Alo if A Solution In Example 4.21, we found that this matrix has eigenvalues A, = -1 and 12 = 2, with corresponding eigenvectors v, = _ and v2 = 2 . It follows (from any one of a number of theorems in this section) that A is diagonalizable and PAP = D, where P = [v, val = _ _ and D = [ - 2 Solving for A, we have A = PDP- and, by Theorem 4.22(f), A" = PD"P-' for all n 21. Since D" = - 2- we have A" = PD"P-1 = 2(-1)" + 2" (-1)"+ 1 + 2" = 3 2 (-1) "+ 1 + 2 " + 1 (-1) "+2 + 2"+1 Since we were only asked for A , this is more than we needed. But now we can simply set n = 10 to find [2(-1)10 + 210 (-1)"1 + 210 A = 342 341 2(-1)"1 + 211 (-1)12 + 211 682 683 3