HEY CAN YOU HELP ME SOLVE THE FOLLOWING QUESTIONS: 63, 65, 73, 81, 82, 83

5.3 Diagonalization of Matrices 323 57. [_7 3]: ([:] . ): 43 74. Find a 2 x 2 matrix having eigenvalues 7 and -4, with 58. 2 1]: 1. ): -2 -3 corresponding eigenvectors -3 and [_1]- 75. Find a 3 x 3 matrix having eigenvalues 3, -2, and 1, with 59. [_$ 8]: {[-3] [ ]): 23 corresponding eigenvectors 76. Find a 3 x 3 matrix having eigenvalues 3, 2, and 2, with 61. 9 9 : . 8 :5.5.1 corresponding eigenvectors 77. Give an example of diagonalizable n x n matrices A and B such that A + B is not diagonalizable. 62 . 78. Give an example of diagonalizable n x n matrices A and B such that AB is not diagonalizable. In Exercises 63-72, a matrix and its characteristic polynomial 79. Show that every diagonal n x n matrix is diagonalizable. are given. Determine all values of the scalar c for which each 80. (a) Let A be an n x n matrix having a single eigenvalue matrix is not diagonalizable. c. Show that if A is diagonalizable, then A = cln. (b) Use (a) to explain why ( 2 is not diagonalizable. AN 63. C 64 . 3 0 81. If A is a diagonalizable matrix, prove that A is diagonal- -(1 - c)(1 -2)(1 - 3) -(t - c)(1 + 3)(1+ 2) izable. 0 82. If A is an invertible matrix that is diagonalizable, prove 65. 1-1 66. c - 4 that A is diagonalizable. 83. If A is a diagonalizable matrix, prove that A is diagonal- -(1 - c)(12 +7) -(1 - c)(t - 2)(1 -4) izable. -1 84. If A is a diagonalizable matrix, prove that A* is diagonal- 6 OWN O 67. 0 68. - 2 1 izable for any positive integer k. 85. Suppose that A and B are similar matrices such that -(t - c)(t - 3)(1 -4) -(1 - c)(12 +8) B = P-AP for some invertible matrix P. -3 0 (a) Show that A is diagonalizable if and only if B is diag- -6 c - 2 0 onalizable. 69. 10 0 70. (b) How are the eigenvalues of A related to the eigenval- -(1 - c)(t + 2)(1 + 1) -(t - c)(t + 2)(1 - 3) ues of B? Justify your answer. (c) How are the eigenvectors of A related to the eigen- -9 -3 -15 -7 0 rectors of B? Justify your answer. -6 71 7 2 13 86. A matrix B is called a cube root of a matrix A if 4 0 3 B3 = A. Prove that every diagonalizable matrix has a cube root. (t - c)(1 + 3 ) (1 + 1)(1 -2) 87. Prove that if a nilpotent matrix is diagonalizable, then C 6 2 10 it must be the zero matrix. Hint: Use Exercise 72 of -12 0 -15 Section 5.1. 72 -11 1 -15 10 0 13 88. Let A be a diagonalizable n x n matrix. Prove that if the characteristic polynomial of A is f(1) = (t - c)(1 + 2)(t - 1)(1 -3) ant" + an-11" - + ... + ait + do, then f(A) = 0, where 73. Find a 2 x 2 matrix having eigenvalues -3 and 5, with f(A) = anA" + an-14"- + ... + ajA + don. (This corresponding eigenvectors |and - result is called the Cayley-Hamilton theorem.?) Hint: If A = PDP-, show that f(A) = Pf(D )P-1