962 CHAPTER 9 Systems and Matrices For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. 79. A = c 3 -2 4 1d , B = c 6 5 0 -2d 80. A = c 0 -4 -5 2d , B = c 3 -5 -1 4d 81. A = £ 0 0 0 1 1 0 -1 0 1§, B = £ 1 0 0 0 1 0 0 0 1§ 82. A = £ -1 0 -1 0 1 -1 1 1 0§, B = £ 0 0 1 0 1 0 1 0 0§ 83. Concept Check In Exercise 81, AB = A and BA = A. For this pair of matrices, B acts the same way for matrix multiplication as the number acts for multiplication of real numbers. 84. Concept Check Find AB and BA for the following matrices. A = c a c b dd and B = c 1 0 0 1d Matrix B acts as the multiplicative element for 2 * 2 square matrices. Solve each problem. See Example 8. 85. Income from Yogurt Yagel’s Yogurt sells three types of yogurt: nonfat, regular, and super creamy, at three locations. Location I sells 50 gal of nonfat, 100 gal of regular, and 30 gal of super creamy each day. Location II sells 10 gal of nonfat, and Location III sells 60 gal of nonfat each day. Daily sales of regular yogurt are 90 gal at Location II and 120 gal at Location III. At Location II, 50 gal of super creamy are sold each day, and 40 gal of super creamy are sold each day at Location III. (a) Write a 3 * 3 matrix that shows the sales figures for the three locations, with the rows representing the three locations. (b) The incomes per gallon for nonfat, regular, and super creamy are $12, $10, and $15, respectively. Write a 1 * 3 or 3 * 1 matrix displaying the incomes. (c) Find a matrix product that gives the daily income at each of the three locations. (d) What is Yagel’s Yogurt’s total daily income from the three locations? 86. Purchasing Costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A. Eggs Flour Sugar Shortening Milk Rolls (doz) Bread (loaf ) Cake Pie (crust) E 1 0 4 0 4 3 3 1 1 4 0 2 0 1 4 1 4 1 1 3 1 0 1 0U = A The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix B. Cost Large Lot Small Lot Eggs Flour Sugar Shortening Milk E 5 8 10 12 5 5 10 12 15 6U = B (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options.
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