9.4 Matrices 579 In Exercises 47–50, show that the associative property of multiplication, A B C A B C ( ) ( ), × × = × × holds for matrices A B, , and C. 47. A B C 1 3 4 0 , 4 2 3 1 , 2 1 3 0 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 48. A B C 2 3 0 4 , 4 0 3 5 , 3 4 2 5 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 49. A B C 4 3 6 2 , 1 2 0 1 , 4 3 0 2 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 50. A B C 3 4 1 2 , 0 1 1 0 , 2 0 3 0 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * Problem Solving 51. MODELING—High School Play North Shore High School sold tickets for the musical The Lion King. Matrix A indicates the number of student, adult, and senior citizen tickets sold for the matinee and evening shows on day 1. Matrix B indicates the number of student, adult, and senior citizen tickets sold for the matinee and evening shows on day 2. Use matrix subtraction to determine the number of T-shirts sold in the given day for each size of each type. * 53. MODELING—Cookie Company Costs Kasper’s Kookies bakes and sells four types of cookies: chocolate chip, sugar, molasses, and peanut butter. Matrix A indicates the number of units of various ingredients used in baking a dozen of each type of cookie. A 2 2 1 3 2 1 2 0 1 0 3 1 0 0 1 2 1 2 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ Student Adult Senior citizen A 85 150 50 95 162 41 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Matinee Evening Student Adult Senior citizen B 73 130 45 120 200 53 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Matinee Evening Use matrix addition to determine the total number of student, adult, and senior citizen tickets sold by North Shore High School for both days for the matinee and evening shows. * 52. MODELING—T-shirt Inventory Hollister sells two types of T-shirts—men’s and women’s. Matrix A indicates the stock on hand of each type and size of T-shirt at the beginning of a given day. Matrix B indicates the stock on hand of each type and size of T-shirt at the end of the same given day. A 31 18 39 16 41 22 34 21 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ B 14 9 18 9 19 15 15 9 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Douglas Graham/Loudoun Now/William Graham/Alamy Stock Photo Men’s Women’s Small Medium Large Extra large Men’s Women’s Small Medium Large Extra large Sugar Flour Milk Eggs Chocolate chip Sugar Molasses Peanut butter The cost, in cents per cup or per egg, for each ingredient when purchased in small quantities and in large quantities is indicated in matrix B below. B 10 12 5 8 8 8 4 6 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Large quantities Small quantities Sugar Flour Milk Eggs Use matrix multiplication to determine a matrix representing the comparative cost per item for small and large quantities purchased. * In Exercises 54 and 55, use the information given in Exercise 53. Suppose a typical day’s order consists of 40 dozen chocolate chip cookies, 30 dozen sugar cookies, 12 dozen molasses cookies, and 20 dozen peanut butter cookies. 54. Cookie Orders a) Express these orders as a 1 4 × matrix. ⎡⎣ ⎤⎦ 40 30 12 20 b) Use matrix multiplication to determine the amount of each ingredient needed to fill the day’s order. 180 172 50 136 ⎡⎣ ⎤⎦ 55. Cookie Costs Use matrix multiplication to determine the cost, in dollars, under the two purchase options (small and large quantities) to fill the day’s order. ⎡⎣ ⎤⎦ 36.04 47.52 56. MODELING—Food Prices To raise money for a local community organization, the Spanish Club at Riverhead High School sold hot dogs, soft drinks, and candy bars for 3 days in the student lounge. The sales for the 3 days are summarized in matrix A. A 52 50 75 48 43 60 62 57 81 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ Day 1 Day 2 Day 3 Hot dogs Soft drinks Candy bars The cost and revenue (in dollars) for hot dogs, soft drinks, and candy are summarized in matrix B. *See Instructor Answer Appendix
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