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C-96 Appendix C Concluding Remarks 1. As you can see after working this exercise, for larger networks, it becomes preferable to work from the matrices rather than a diagram. For instance, what if there were, say, 50 spies, rather than only 4 or 5? 2. The mathematics in this project can be applied in many other areas. For instance, instead of communication between spies, the subject could be air travel between cities. Then, for example, a one-intermediary route from New York to San Francisco would correspond to a flight between those two cities with one stopover, for example, in Chicago.
Appendix C C-97 PROJECT The Leontief Model Revisited As background here, you need to be familiar with the project beginning on page C-91 about the Leontief input-output model. Additionally, you need to have studied Example 2 in the present section, where matrix methods are applied to solve a system of equations. If necessary, one or two groups can be assigned to work through or to review that material and then present it to the class at large. In this project, you’ll use matrix methods to shorten the work in solving an input-output problem. PART I: How to solve the matrix equation AX D X In Example 2 of Section 11.4,* we solved the matrix equation AX B. Along the same lines, now we want to solve the matrix equation AX D X (1) (We assume that X is a square matrix with an inverse and that each matrix has the appropriate numbers of rows and columns for the calculations to make sense.) We start by subtracting X from both sides of equation (1) to obtain AX D X 0 AX X D (A I)X D Recall from Section 11.3 that 0 denotes the zero matrix, in which every entry is the number 0. subtracting D from both sides and using the property 0 D D I denotes the identity matrix. We’ve used the fact that the distributive property is valid in matrix algebra just as it is in ordinary algebra. To isolate X, assuming A I is invertible, multiply both sides of this last equation by (A I) 1 . This yields (A I) 1 (A I)X (A I) 1 (D) IX (A I) 1 (D) X (A I) 1 (D) using IX X (2) Equation (2) is the required solution. To see how this works in a simple case, suppose we want to solve equation (1), given that A a 3 5 D a 10 and X a x 1 4 b , 20 b , That is, the equation to be solved is y b . a 3 5 1 4 bax y b a 10 20 b a x y b (3) *Precalculus: A Problems-Oriented Approach, 7th edition
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APPENDIX C Title and Page Number Pr
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Appendix C C-3 MINI PROJECT Discuss
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Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
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Appendix C C-9 MINI PROJECT Flying
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Appendix C C-11 MINI PROJECT An Ine
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Appendix C C-13 (a) The two data po
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Appendix C C-15 and http://www.svob
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Appendix C C-17 For Figure D we can
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Appendix C C-19 MINI PROJECT A Grap
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Appendix C C-21 MINI PROJECT Who Ar
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Appendix C C-23 MINI PROJECT 1 How
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Appendix C C-25 either the quadrati
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Appendix C C-27 PROJECT The Least-S
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Appendix C C-29 Then, for any line
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Appendix C C-31 PROJECT Finding Som
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Appendix C C-33 3. When a person co
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Appendix C C-35 2. Use part (b) of
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Appendix C C-37 PROJECT Coffee Temp
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Appendix C C-39 MINI PROJECT More C
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Appendix C C-41 The calculations co
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Appendix C C-43 PROJECT Transits of
Page 45 and 46: Appendix C C-45 E A ¨ V ¨ Bª d S
Page 47 and 48: Appendix C C-47 PROJECT A Linear Ap
Page 49 and 50: Appendix C C-49 Continuing with the
Page 51 and 52: Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
Page 61 and 62: Appendix C C-61 PROJECT The Design
Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
Page 67 and 68: Appendix C C-67 PROJECT Inverse Sec
Page 69 and 70: Appendix C C-69 Alternatively, usin
Page 71 and 72: Appendix C C-71 PROJECT Snell’s L
Page 73 and 74: Appendix C C-73 vertically downward
Page 75 and 76: 0 0 0 Appendix C C-75 PROJECT Vecto
Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
Page 81 and 82: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
Page 83 and 84: Appendix C C-83 PROJECT Parameteriz
Page 85 and 86: Appendix C C-85 It is worth noting
Page 87 and 88: Appendix C C-87 6. Show that the li
Page 89 and 90: Appendix C C-89 2. An altitude of a
Page 91 and 92: Appendix C C-91 PROJECT The Leontie
Page 93 and 94: Appendix C C-93 n x units of that
Page 95: Appendix C C-95 him/herself through
Page 99 and 100: Appendix C C-99 As in the solution
Page 101 and 102: Appendix C C-101 MINI PROJECT 2 Con
Page 103 and 104: Appendix C C-103 PROJECT Using Hype
Page 105 and 106: Appendix C C-105 PROJECT Two Method
Page 107 and 108: Appendix C C-107 (b) Use Method 1 t
Page 109 and 110: Appendix C C-109 MINI PROJECT An Un
Page 111 and 112: Appendix C C-111 PROJECT More on Su
Page 113 and 114: Appendix C C-113 SOLUTION 25 25 a (
Page 115: Appendix C C-115 4. Simplify the fo
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