Name: Unit 4 Matrix Methods Lesson 1 PRACTICE PROBLEMS Constructing, Interpreting, and Operating on Matrices I can construct matrices to organize, display, and analyze information, interpret those matrices, and carry out matrix operations. Investigation Practice Problem Options Max Possible Points Investigation 1: There s No Business Like Shoe Business #2, 7 6 points Total Points Earned Investigation 2: Analyzing Matrices #1 6 points Investigation 3: Combining Matrices #3, 4, 5, 6 20 points /32 points ** In order to earn credit for practice problems, ALL WORK must be shown.**
Applications 1 The movie matrix from Investigation 2 is reproduced below. Recall that the movie matrix can be thought of as describing friendship, and it is read from row to column. Thus, for example, student A considers student B as a friend since there is a 1 in the first row and second column. /3 /3 Movie Matrix Would Like to Go to a Movie with A B C D E A 0 1 1 1 1 B 0 0 1 1 1 C 1 0 0 1 0 D 0 1 1 0 1 E 1 0 0 1 0 a. Mutual friends are two people who consider each other as friends. i. Are students A and D mutual friends? ii. Find at least two pairs of mutual friends. iii. How do mutual friends appear in the matrix? b. Construct a new matrix that shows mutual friends. To do this, list all five people across the top and also down the side. Write a 1 or a 0 for each entry, depending on whether or not the two people corresponding to that entry are mutual friends. c. Who has the most mutual friends? d. Compare the first row of the mutual-friends matrix to the first column. Compare each of the other rows to its corresponding column. What relationship do you see? Explain why the mutual-friends matrix has this relationship between its rows and columns. (See Connections Task 8 for more information about matrices with this property.) 2 Spreadsheets are one of the most widely-used software applications. A spreadsheet displays organized information in the same way a matrix does. One common use of spreadsheets is to itemize loans. For example, suppose that you are going to buy your first car. The one you decide to buy needs some work, but you can get it for $500. You borrow the $500 at 9% annual interest and agree to pay it back in 12 monthly payments. The following spreadsheet summarizes all the information about this loan. LESSON 1 Constructing, Interpreting, and Operating On Matrices 87
= = a. How much principal will you still owe after the sixth payment? b. How much interest will you pay in the fourth month? c. In any given row, how do the entries in the To Interest and To Principal columns compare to the entry in the Payment column? Why are the entries related in this way? d. Why do the entries in the To Principal column get bigger throughout the loan period? e. How can you use nearby entries to compute the entries in the Month 10 row? f. How much money will you save if you pay for the car in cash instead of borrowing the $500 and paying off the loan over a year? 3 Music is an enjoyable part of many people s lives. The format in which you get music has changed over the years and continues to change. The following matrices show shipments of music albums in the United States in different formats during the transition period between the 1990s and 2000s. The formats shown are: CD compact disc; CASS cassette; VNL vinyl; DL download. The 0* in the DL column means that it was not possible to download music at that time, or data were not yet being gathered for this format, or the number is so small that it rounds to 0. The numbers shown are millions of units. 88 UNIT 2 Matrix Methods
Album Shipments Late 1990s (in millions) CD CASS VNL DL 1994 662.1 345.4 1.9 0* 1995 722.9 272.6 2.2 0* 1996 778.9 225.3 2.9 0* 1997 753.1 172.6 2.7 0* 1998 847.0 158.5 3.4 0* 1999 938.9 123.6 2.9 0* Album Shipments Early 2000s (in millions) CD CASS VNL DL 2000 942.5 76.0 2.2 0* 2001 881.9 45.0 2.3 0* 2002 803.3 31.1 1.7 0* 2003 746.0 17.2 1.5 0* 2004 767.0 5.2 1.36 4.6 2005 705.4 2.5 1.02 13.6 On Your Own Source: Recording Industry Association of America; www.riaa.com/news/marketingdata/facts.asp a. Describe any patterns you see in these data. b. Analyze these matrices. Use the rows or columns of the matrices to help answer the following questions. i. How many millions of albums were shipped in all 4 formats in 2005? ii. How many CDs were shipped in the early 2000s? iii. How many more cassettes were shipped in the late 1990s than in the early 2000s? c. Do these two matrices have appropriate sizes so they could be added? Would it make sense to add these matrices? Explain. d. Construct a 2 4 matrix that shows the total number of album shipments in each of the 4 formats for the late 1990s and the early 2000s. Use the matrix to help answer the following questions. i. Which format shows the greatest number of increased shipments from the late 1990s to the early 2000s? Which format shows the most dramatic increase in shipments from the late 1990s to the early 2000s? ii. Consider the total albums shipped in all 4 formats. Which period, the late 1990s or the early 2000s, had more albums shipped? How many more? iii. Describe the general trends shown in this matrix. Do you think these trends will continue? Explain. 4 An automotive manufacturer produces several styles of sport wheels. One of the styles is available in two finishes (chrome-plated and silver-painted) and three wheel sizes (15-inch, 16-inch, and 17-inch). In October, a retailer in the Midwest purchased sixteen 15-inch chrome wheels, twenty-four 16-inch chrome wheels, eight 17-inch chrome wheels, eight 15-inch silver wheels, twelve 16-inch silver wheels, and four 17-inch silver wheels. In November, the retailer ordered twelve 15-inch chrome wheels, thirty-two 16-inch chrome wheels, sixteen 17-inch chrome wheels, twelve 15-inch silver wheels, and twenty 16-inch silver wheels. a. Construct two matrices that show the wheel orders one for October and one for November. Label the matrices and the rows and columns. LESSON 1 Constructing, Interpreting, and Operating On Matrices 89
b. How many of each type of wheel were ordered by the retailer during these two months combined? Represent this information in a matrix. Label the matrix and the rows and columns. c. Suppose that over the entire fourth quarter (October, November, and December) the retailer has agreed to order the number of wheels shown in the following matrix. Fourth-Quarter Order 15-in. 16-in. 17-in. Chrome 40 52 36 Silver 28 32 16 i. Construct a matrix that shows how many of each type of wheel must be ordered in December to meet this agreement. ii. Explain any unusual entries in the matrix. d. In October of the next year, the retailer orders twice the number of each type of wheel ordered the previous October. November s order is three times the number of each type of wheel ordered the previous November. Construct a matrix that shows the number of each type of wheel ordered in the two months combined. 5 The first matrix below presents combined monthly sales for three types of men s and women s jeans at JustJeans stores in three cities. The second matrix below gives the monthly sales for women s jeans. Combined Sales Levi Lee Wrangler Chicago 250 195 105 Atlanta 175 175 90 San Diego 185 210 275 Women s Jeans Sales Levi Lee Wrangler Chicago 100 90 70 Atlanta 80 85 50 San Diego 105 50 150 a. Construct a matrix that shows the monthly sales for men s jeans for each brand and each city. Which matrix operation did you use to construct this matrix? 90 UNIT 2 Matrix Methods
/3 b. Organizing the data in different ways can highlight different information. Copy and complete the following matrices to show sales of men s and women s jeans in each city, for each of the three brands. Label the rows. Chicago M W Atlanta M W San Diego M W On Your Own c. Refer to the matrices in Part b. Construct one matrix that shows the total sales of men s and women s jeans for each of the three brands, that is, sales in all three cities combined. Label the rows of the matrix with the brands and the columns with M and W. Which matrix operation did you use to construct this matrix? d. For the first quarter, the managers of the Chicago, Atlanta, and San Diego stores have placed jeans orders with the main warehouse as indicated in the matrices below. Let C represent the matrix for the store in Chicago, A for the store in Atlanta, and S for the store in San Diego. Chicago M W Levi 300 330 Lee 345 300 Wrangler 120 240 Atlanta M W 300 255 300 270 135 165 San Diego M W 252 315 513 162 405 450 For the second quarter, the managers orders for each brand are tripled in Chicago, stay the same in Atlanta, and are _ 2 as big in 3 San Diego. i. Think about how to calculate the total-orders matrix T of men s and women s jeans in all three cities combined, for each of the three brands for the second quarter. Write a rule for calculating T in terms of C, A, and S. ii. Compute the second row, second column entry of T. What does this entry tell you about jeans orders placed with the warehouse? 6 Consider the following matrices. A = 2-4 6 0 1.5 3 7-3.5 8 1-1 6 B = 2 3-6 0 1 6.5 11-3 6 C = 1 0 0 1 1 1 a. Compute B + D. b. Compute 6C. D = -1 1.25 0 8-12 5 0 0 18 LESSON 1 Constructing, Interpreting, and Operating On Matrices 91
c. Compute -A. d. Compute B + B. e. Compute 2B - 3D. f. Compute D - B. g. Construct a new matrix E that could be added to A. Then compute A + E. Connections 7 In Problem 4 of Investigation 1, you examined some monthly shoe sales data, reproduced below. Each entry represents the number of pairs of shoes sold. Monthly Sales J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 35 40 70 a. What is the mean number of pairs of Reeboks sold per month? b. Which brand has more variability in its monthly sales? Explain how you determined variability. c. Identify at least two types of data plots that could be used to represent the monthly sales data. d. Create a plot that you think would be most informative. 8 Symmetry is an important concept in mathematics. In prior units of Core-Plus Mathematics, you examined geometric shapes and graphs of functions in terms of their symmetries. Symmetry also applies to matrices, but only to square matrices. A square matrix is said to be symmetric if it has reflection (or mirror) symmetry about its main diagonal. (Recall that the main diagonal of a square matrix is the diagonal line of entries running from the top-left to the bottom-right corner.) So, a square matrix is symmetric if the numbers in the mirror-image positions, reflected in the main diagonal, are the same. For example, consider the three matrices below. Matrices A and B are symmetric, but matrix C is not symmetric. A = 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 B = 25 3 4 5 3 36 6 7 4 6 9 8 5 7 8 10 C = 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 0 a. Identify two square matrices from this lesson. 92 UNIT 2 Matrix Methods