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Download 2-3 Using Matrices to Model Real-World Data
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Advanced Mathematical Concepts Chapter 2 Lesson 2-3 Example 1 DESSERT Jessica does a survey on the cost of four different desserts at three local restaurants. At restaurant A, a slice of apple pie is priced at $2.25, a brownie sundae is priced at $2.95, a slice of apple cobbler is priced at $1.95, and an ice cream cone is priced at $1.10. At restaurant B, apple pie is $2.75, a brownie sundae is $3.45, apple cobbler is $2.50, and an ice cream cone is $1.65. At restaurant C, apple pie is $2.40, a brownie sundae is $2.70, apple cobbler is $2.35, and an ice cream cone is $1.15. a. Use a matrix to represent the data. b. Use a symbol to represent the price of a brownie sundae at restaurant C. a. To represent data using a matrix, choose which category will be represented by the columns and which will be represented by the rows. Let’s use the columns to represent the prices at each restaurant and the rows to represent the prices of each dessert. Then write each data piece as you would if you were placing the data in a table. apple pie brownie sundae apple cobbler ice cream A $2.25 $2.95 $1.95 $1.10 B $2.75 $3.45 $2.50 $1.65 C $2.40 $2.70 $2.35 $1.15 Notice that the category names appear outside of the matrix. b. The price of a brownie sundae at restaurant C is found in row 2, column 3 of the matrix. This element is represented by the symbol a23. Example 2 y 3 x + 16 Find the values of x and y for which the matrix equation is true. x 3 y Since the corresponding elements are equal, we can express the equality of the matrices as two equations. y = 3x + 16 x = 3y Solve the system of equations by using substitution. y = 3x + 16 y = 3(3y) + 16 y = -2 Substitute 3y for x. Solve for y. x = 3(-2) x = -6 Substitute –2 for y in the second equation to find x. The matrices are equal if x = -6 and y = -2. Check by substituting into the matrices. Advanced Mathematical Concepts Chapter 2 Example 3 4 -2 6 Find A + B if A = and B = 1 3 -3 -1 2 5 -4 1 7 . 4 (1) 2 2 6 5 A+B = 1 (4) 3 1 3 7 3 0 11 = 3 4 4 Example 4 5 2 8 1 and D = Find C – D if C = -4 3 2 -1 2 5 -3 4 . 6 -8 3 5 C – D = C + -D 5 2 2 5 8 1 3 4 = 4 3 6 8 2 1 3 5 5 (2) 2 (5) 3 83 11 1 (4) or = 4 (6) -10 38 2 (3) 1 (5) -1 -3 -3 11 -6 Advanced Mathematical Concepts Chapter 2 Example 5 1 3 4 If A = 2 5 0 , find 2A. 3 6 2 1 3 4 2 2 5 0 = 3 6 2 2(1) 2(2) 2(3) 2 6 = 4 10 6 12 2(3) 2(4) 2(5) 2(0) 2(6) 2(2) 8 0 4 Multiply each element by 2. Example 6 2 4 Use matrices A = ,B= 0 1 3 1 -2 -3 4 2 4 0 -1 , and C = 1 5 0 to find each product. a. AB 2 4 3 1 2 AB = 0 1 4 0 1 2(3) 4(4) 2(1) 4(0) 2(2) 4(1) 22 2 -8 or AB = 0(3) 1(4) 0(1) 1(0) 0(2) 1(1) 4 0 -1 b. BC B is a 2 3 matrix and C is a 2 3 matrix. Since B does not have the same number of columns as C has rows, the product BC does not exist. BC is undefined. Advanced Mathematical Concepts Example 7 SHOPPING At a certain clothing store, each pair of jeans (J) is priced at $15, each t-shirt (T) is priced at $10, and each sweater (S) is priced at $20. The chart lists the number of each of these items purchased by five shoppers. Use matrix multiplication to find the total amount spent by each shopper. Chapter 2 Shopper Sarah Dave Jessica Drew Emily Jeans 1 2 3 0 1 T-Shirts 3 2 1 4 2 Sweaters 1 0 2 1 2 Write the purchase information as a 5 3 matrix and write the prices as a 3 1 matrix. Then multiply the matrices. J T S Sarah Dave Jessica Drew Emily 1 2 3 0 1 3 2 1 4 2 Cost 1 J 15 0 T 10 2 S 20 1 2 = Cost Sarah Dave Jessica Drew Emily 1(15) + 3(10) + 1(20) 2(15) + 2(10) + 0(20) 3(15) + 1(10) + 2(20) 0(15) + 4(10) + 1(20) 1(15) + 2(10) + 2(20) Cost = Sarah Dave Jessica Drew Emily 65 50 95 60 75
