976 CHAPTER 9 Systems and Matrices 57. Social Security Numbers It is possible to find a polynomial that goes through a given set of points in the plane by using a process called polynomial interpolation. Recall that three points define a second-degree polynomial, four points define a third-degree polynomial, and so on. The only restriction on the points, because polynomials define functions, is that no two distinct points can have the same x-coordinate. Using the SSN 539-58-0954, we can find an eighth-degree polynomial that lies on the nine points with x-coordinates 1 through 9 and y-coordinates that are digits of the SSN: 11, 52, 12, 32, 13, 92, c , 19, 42. This is done by writing a system of nine equations with nine variables, which is then solved by the inverse matrix method. The graph of this polynomial is shown. Find such a polynomial using your own SSN. 58. Repeat Exercise 57 but use -1, -2, c , -9 for the x-coordinates. Use a graphing calculator to find the inverse of each matrix. Give as many decimal places as the calculator shows. See Example 2. −2 −25 15 10 59. c 2 3 22 0.723d 60. c 22 -17 0.5 1 2 d 61. D 1 2 0 1 2 1 4 1 4 1 2 1 3 1 3 1 3T 62. £ 1.4 0.84 0.56 0.5 1.36 0.47 0.59 0.62 1.3 § Use a graphing calculator and the method of matrix inverses to solve each system. Give as many decimal places as the calculator shows. See Example 4. 63. 2.1x + y = 25 22x - 2y = 5 64. x - 22y = 2.6 0.75x + y = -7 65. 1log 22x + 1ln 32y + 1ln 42z = 1 1ln 32x + 1log 22y + 1ln 82z = 5 1log 122x + 1ln 42y + 1ln 82z = 9 66. px + ey + 22z = 1 ex + py + 22z = 2 22x + ey + pz = 3 Let A = c a c b dd , and let O be the 2 * 2 zero matrix. Show that each statement is true. 67. A # O= O # A = O 68. For square matrices A and B of the same dimension, if AB = O and if A-1 exists, then B = O. Work each problem. 69. Prove that any square matrix has no more than one inverse. 70. Give an example of two matrices A and B, where 1AB2-1 ≠A-1B-1. 71. Suppose A and B are matrices, where A-1, B-1, and AB all exist. Show that 1AB2-1 = B-1A-1. 72. Let A = £ a 0 0 0 b 0 0 0 c§, where a, b, and c are nonzero real numbers. Find A-1. 73. Let A = £ 1 0 0 0 0 1 0 -1 -1§. Show that A3 = I 3, and use this result to find the inverse of A. 74. What are the inverses of In, -A (in terms of A), and kA (k a scalar)?
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