SECTION 11.4 Matrix Algebra 811 (c) Each entry in the result for part (b) represents the position of a letter in the English alphabet ( = A 1, = = B C 2, 3, and so on).What is the original message? Source: goldenmuseum.com Use the following discussion for Problems 92 and 93. In graph theory, an adjacency matrix, A, is a way of representing which nodes (or vertices) are connected. For a simple directed graph, each entry, a , ij is either 1 (if a direct path exists from node i to node j) or 0 (if no direct path exists from node i to node j). For example, consider the following graph and corresponding adjacency matrix. 4 2 1 3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 The entry a14 is 1 because a direct path exists from node 1 to node 4. However, the entry a41 is 0 because no path exists from node 4 to node 1.The entry a33 is 1 because a direct path exists from node 3 to itself.The matrix = + + + B A A A k k 2 indicates the number of ways to get from node i to node j within k moves (steps). 92. Website Map A content map can be used to show how different pages on a website are connected. For example, the following content map shows the relationship among the five pages of a certain website with links between pages represented by arrows. Page 1 Page 3 Page 4 Page 2 Page 5 The content map can be represented by a 5 by 5 adjacency matrix where each entry, a , ij is either 1 (if a link exists from page i to page j) or 0 (if no link exists from page i to page j). (a) Write the 5 by 5 adjacency matrix that represents the given content map. (b) Explain the significance of the entries on the main diagonal in your result from part (a). (c) Find and interpret A .2 93. Three-Click Rule An unofficial, and often contested, guideline for website design is to make all website content available to a user within three clicks.The webpage adjacency matrix for a certain website is given by = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ A 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 (a) Find B .3 Does this website satisfy the Three-Click Rule? (b) Which page can be reached the most number of ways from page 1 within three clicks? 89. Computing the Cost of Production The Acme Steel Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10-gallon (gal) capacity, 350 with 5-gal capacity, and 400 with 1-gal capacity. On the same day, the following aluminum containers were manufactured: 700 with 10-gal capacity, 500 with 5-gal capacity, and 850 with 1-gal capacity. (a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data. (b) If the amount of material used in the 10-gal containers is 15 pounds (lb), the amount used in the 5-gal containers is 8 lb, and the amount used in the 1-gal containers is 3 lb, find a 3 by 1 matrix representing the amount of material used. (c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day’s usage of material. (d) If stainless steel costs Acme $0.10 per pound and aluminum costs $0.05 per pound, find a 1 by 2 matrix representing cost. (e) Multiply the matrices found in parts (c) and (d) to find the total cost of the day’s production. 90. Computing Profit Rizza’s Used Cars has two locations, one in the city and the other in the suburbs. In January, the city location sold 400 subcompacts, 250 intermediate-size cars, and 50 SUVs; in February, it sold 350 subcompacts, 100 intermediates, and 30 SUVs. At the suburban location in January, 450 subcompacts, 200 intermediates, and 140 SUVs were sold. In February, the suburban location sold 350 subcompacts, 300 intermediates, and 100 SUVs. (a) Find 2 by 3 matrices that summarize the sales data for each location for January and February (one matrix for each month). (b) Use matrix addition to obtain total sales for the 2-month period. (c) The profit on each kind of car is $100 per subcompact, $150 per intermediate, and $200 per SUV. Find a 3 by 1 matrix representing this profit. (d) Multiply the matrices found in parts (b) and (c) to get a 2 by 1 matrix showing the profit at each location. 91. Cryptography One method of encryption is to use a matrix to encrypt the message and then use the corresponding inverse matrix to decode the message.The encrypted matrix, E, is obtained by multiplying the message matrix, M, by a key matrix, K. The original message can be retrieved by multiplying the encrypted matrix by the inverse of the key matrix. That is, E M K = ⋅ and M E K .1 = ⋅ − (a) The key matrix K 2 1 1 1 1 0 1 1 1 . = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Find its inverse, K .1− [Note: This key matrix is known as the Q2 3 Fibonacci encryption matrix.] (b) Use the result from part (a) to decode the encrypted matrix E 47 34 33 44 36 27 47 41 20 . = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥
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