Applications 1 a. i. No, students A and D are not mutual friends because D does not consider A a friend. ii. The following are the pairs of mutual friends: A-C, A-E, B-D, C-D, and D-E. iii. Each person has to have a 1 for the other person. For example, A will say C is a friend and C will say A is a friend. b. Mutual Friends A B C D E A 0 0 1 0 1 B 0 0 0 1 0 C 1 0 0 1 0 D 0 1 1 0 1 E 1 0 0 1 0 c. Student D has the most mutual friends. d. The entries are the same for each row and corresponding column. The 1s are symmetric about the main diagonal of 0s. Mutual friends each have a 1 for the other person. 2 Spreadsheets are a common and powerful technology tool. You may decide to ask students to create the spreadsheet shown using software such as Excel, then answer the questions. TECHNOLOGY NOTE There are many Internet sites for producing amortization schedules. (Do a search.) Students may notice that different sites produce slightly different schedules, likely because of rounding procedures. CPMP-Tools Constructing, Interpreting, and Operating on Matrices T87
a. $255.58 b. $2.84 c. The sum of To Interest and To Principal is the amount under Payment. The payment is split between paying off interest and paying off the principal. None of it goes to pay anything else. d. The interest is computed on a smaller remaining balance each month, so the interest that is added to the principal is less each month, resulting in more of the payment going toward the principal. e. Cell B13 is the same as B12. Cell C13 is =E12*0.09/12. Cell D13 is =B13-C13. Cell E13 is =E12-D13. f. The interest $24.71 would be saved. 3 INSTRUCTIONAL NOTE You might ask your students to find the latest data about album shipments and use those data in addition to or instead of the data presented here. A good resource Web site is the site for the Recording Industry Association of America: www.riaa.com. Constructing, Interpreting, and Operating on Matrices T88
a. Some patterns: Although there are only two years of data for the download format, this format seems to be growing very strongly. Cassettes are decreasing to apparent extinction. Vinyl is small but seems to be surviving at a low level. CDs seem to still be popular but declining. b. i. The sum of the 2005 row is 722.52. Thus, 722.52 millions of albums were shipped in all 4 formats in 2005. ii. The sum of the CD column in the Early 2000s matrix is 4,846.1. Thus, 4,846,100,000 CDs were shipped in the early 2000s (2000 2005). iii. Subtract the Early 2000s CASS column sum from the Late 1990s CASS column sum. 1,121,000,000 more cassettes were shipped in the late 1990s than in the early 2000s. c. These matrices do have the same size, so they could be added. However, the resulting data would not make much sense. d. Album Shipments in Different Formats (in millions) CD CASS VNL DL Late 1990s 4,702.9 1,289 16 0 Early 2000s 4,846.1 177 10.08 18.2 i. The CD format shows the greatest increase in numbers. The DL format shows the most dramatic increase. ii. By computing row sums and then subtracting, we can see that there were 956,520,000 more album shipments in the late 1990s than in the early 2000s. iii. The trends that students describe here may be similar to the trends described in Part bi. Some of those trends are more easily seen in the matrix above: CD shipments did not change much from the late 1990s to the early 2000s; cassette shipments decreased sharply; vinyl shipments are small but not declining as sharply as cassettes; downloads are growing dramatically. 4 This problem begins by asking students to construct a matrix based on information contained in paragraph form. This illustrates again how matrices work well to organize data and are much easier to understand than information in paragraph form. a. October November O = Chrome 16 24 8 Silver 8 12 4 N = Chrome 12 32 16 Silver 12 20 0 Constructing, Interpreting, and Operating on Matrices T89
b. The total ordered during these two months combined is the sum of the two matrices in Part a. October November Order Chrome 28 56 24 Silver 20 32 4 c. i. December Chrome 12-4 12 Silver 8 0 12 ii. The retailer has already met and exceeded the quota for ordering chrome 16-inch wheels. Thus, there is a negative value in the matrix. In reality, the retailer would not be required to order any more chrome 16-inch wheels. d. Calculate (2 O) + (3 N). Chrome 68 144 64 Silver 52 84 8 5 a. This matrix is obtained by subtracting the women s jeans sales matrix from the combined sales matrix. Men s Jeans Sales Levi Lee Wrangler Chicago 150 105 35 Atlanta 95 90 40 San Diego 80 160 125 Constructing, Interpreting, and Operating on Matrices T90
b. Chicago Atlanta San Diego Levi 150 100 Lee 105 90 Wrangler 35 70 Levi 95 80 Lee 90 85 Wrangler 40 50 Levi 80 105 Lee 160 50 Wrangler 125 150 c. You get the following matrix by adding the three matrices above. Combined Sales Levi 325 285 Lee 355 225 Wrangler 200 270 d. i. 3C + A + 2_ 3 S = T ii. 3(300) + 270 + 2_ (162) = 1,278 3 This tells us that for these stores, a total of 1,278 pairs of women s Lee jeans were ordered during the second quarter. 6 a. B + D = b. 6C = 6 0 0 6 6 6 1 4.25-6 8-11 11.5 11-3 24 Constructing, Interpreting, and Operating on Matrices T91
c. -A = -2 4-6 0-1.5-3 -7 3.5-8 -1 1-6 d. B + B = 4 6-12 0 2 13 22-6 12 e. 2B - 3D = 7 2.25-12 -24 38-2 22-6 -42-3 -1.75 6 f. D - B = 8-13 -1.5-11 3 12 g. Answers will vary depending on student choice for matrix E. Note however, that E must be a 4 3 matrix. Connections 7 a. 50.8333 pairs (Find the mean of the row.) b. For these data, Reebok has the most variability. Students should determine the variability of each row and then compare the three variabilities. Possible measures: the range of values, the standard deviation, the interquartile range. Or, students might explain based on an informal analysis and description of a data plot, like box plots, line plots, bar graphs, or stemplots. c. Possible types of data plots: stemplots, plots over time, bar graphs, or box plots. d. Responses may vary. Several graphs could be useful, for example, plots over time or box plots with all three brands on the same graph, pairwise back-to-back stemplots, or combined bar graphs. 8 a. The square matrices include the movie matrix, the loan matrix, the degree-of-difference matrix, and the trustworthy-friend matrix. Constructing, Interpreting, and Operating on Matrices T92