580 CHAPTER 9 Mathematical Systems B 0.35 0.75 0.30 0.70 0.25 1.00 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 61. Create two square matrices, A and B, with the same dimensions and determine whether A B B A. × = × Answers will vary. 62. Create three 2 2 × matrices, A B, , and C, and show that A B C A B C ( ) ( ). × × = × × Answers will vary. Challenge Problems/Group Activities 63. For 2 2 × matrices, the multiplicative identity matrix is I 1 0 0 1 . = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Show, for any 2 2 × matrix, A a b c d , = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ that A I I A A. × = × = a b c d a b c d a b c d 1 0 0 1 1 0 0 1 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 64. Two 2 2 × matrices, A and B, whose product is the multiplicative identity matrix (see Exercise 63) are said to be multiplicative inverses. That is, if A B B A I, × = × = then A and B are multiplicative inverses. In parts (a) and (b) below, show that matrices A and B are multiplicative inverses. a) A B 5 2 2 1 , 1 2 2 5 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5 2 2 1 1 2 2 5 1 2 2 5 5 2 2 1 1 0 0 1 − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ b) A B 7 3 2 1 , 1 3 2 7 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 7 3 2 1 1 3 2 7 1 3 2 7 7 3 2 1 1 0 0 1 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Research Activities 65. Matrices and Systems of Equations In Section 6.7 we solved systems of linear equations algebraically. Systems of linear equations may also be solved using matrix multiplication. Do research and show how the system of equations x y x y 3 2 5 4 3 16 + = − = − can be solved using matrix multiplication. 66. Matrix Display Find an example from a magazine, a newspaper, or the Internet that displays information using a matrix. Write a paper explaining how to interpret the information provided by the matrix. 67. Secret Messages The study of encoding and decoding messages is called cryptography. (See the Recreational Math box on page 562.) Write a paper on how matrices and matrix multiplication are currently used in cryptography. *See Instructor Answer Appendix Cost Revenue Hot dogs Soft drinks Candy bars Multiply the two matrices to form a 3 2 × matrix that shows the total cost and revenue for each item. * 57. Consider the mathematical system consisting of the set of all 3 2 × matrices under the operation of matrix addition. a) Is the system closed? Yes b) Is there an additive identity matrix? If so, what is it? Yes 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ c) Does each matrix in the set have an additive inverse matrix? Yes d) The associative property holds for matrix addition. Give an example. Answers will vary. e) Does the commutative property hold for matrix addition? Yes f) Does this mathematical system form a commutative group? Yes 58. Consider the mathematical system consisting of the set of all 2 4 × matrices under the operation of matrix addition. a) Is the system closed? Yes b) Is there an additive identity matrix? If so, what is it? c) Does each matrix in the set have an additive inverse matrix? Yes d) The associative property holds for matrix addition. Give an example. Answers will vary. e) Does the commutative property hold for matrix addition? Yes f) Does this mathematical system form a commutative group? Yes Concept/Writing Exercises 59. Create two matrices, A and B, with the same dimensions and show that A B B A. + = + Answers will vary. 60. Create three matrices, A B, , and C, with the same dimensions and show that A B C A B C ( ) ( ). + + = + + Answers will vary. Yes 0 0 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥
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