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C-98 Appendix C Using equation (2), the solution is X (A I) 1 (D) ca 3 5 1 4 b a 1 0 0 1 bd 1 a 10 20 b a 2 5 1 3 b 1 a 10 20 b a 3 5 1 2 ba10 20 b a 70 30 b carrying out the matrix subtraction computing the inverse carrying out the matrix multiplication Thus the solution of equation (3) is X a 70 and the values of the unknowns x and y are 70 and 30, 30 b , respectively. Practice Problems for Part I 1. Solve AX D X, given that A a 11 9 D a 25 and X a x 6 5 b , 15 b , y b . 1 3 2 20 2. Solve AX D X, given that A ° 2 7 7 ¢ , D ° 50 ¢ , 1 6 12 70 x X ° y ¢ . z PART II: The Matrix Equation AX D X Applied to an Input-Output Problem As our example, we’ll use the same problem discussed in Part I of the project at the end of Section 11.2.* For ease of reference, we restate the problem here. Given the input-output data in Table 1 and the external demands in Table 2, determine an appropriate production level for each sector so that both the internal and external demands on the economy are satisfied. *Precalculus: A Problems-Oriented Approach, 7th edition TABLE 1 Input-Output Table (internal demands) Outputs Steel Coal Electricity Inputs Steel 0.04 0.02 0.16 Coal 0.12 0 0.30 Electricity 0.16 0.10 0.04 TABLE 2 External Demands External demands Steel 10 units Coal 12 units Electricity 15 units
Appendix C C-99 As in the solution in Section 11.2, we start by using x, y, and z to represent the required number of units of steel, coal, and electricity, respectively. Next, define three matrices, A, D, and X as follows: 0.04 0.02 0.16 10 x A ° 0.12 0 0.30 ¢ D ° 12 ¢ X ° y ¢ (4) 0.16 0.10 0.04 15 z The input-output matrix, The demand The matrix based on Table 1 matrix based of unknowns on Table 2 Your tasks now are to work through the following steps to solve this inputoutput problem using matrix methods. 1. Check that the matrix product AD is defined. What will be the dimension of the product AD? 2. Explain why the matrix equation AX D X holds. Hint: In the project for page C-91 look at the equation immediately preceding equation (2). 3. Use equation (2) from part I of this project to solve the equation AX D X. Make use of a graphing utility in carrying out the matrix operations, and round the answers for x, y, and z to two decimal places. PART III: A Seven-Sector Model of the U.S. Economy The input-output examples above (and in the project on page C-91) involve two- or three-sector economies. By way of contrast, input-output tables for the U.S. economy (published by the U.S. Bureau of Economic Analysis) involve as many as 486 sectors. Table 3, which follows, is an aggregated version of one of those tables, in which the 486 sectors have been grouped into only 7 general categories or sectors. The names of the 7 sectors, abbreviated in Table 3, are spelled out in Table 4, which contains the external demands. Use a graphing utility to solve the input-output problem for this seven-sector economy. Round the final answers to the nearest integer. Then, given that each unit represents one million dollars, express the answers in dollars. TABLE 3 Seven-Sector Input-Output Table for the 1967 U.S. Economy* Outputs Ag Min Con Mfg T & T Serv Oth Inputs Ag .2939 .0000 .0025 .0516 .0009 .0081 .0203 Min .0022 .0504 .0090 .0284 .0002 .0099 .0075 Con .0096 .0229 .0003 .0042 .0085 .0277 .0916 Mfg .1376 .0940 .3637 .3815 .0634 .0896 .1003 T & T .0657 .0296 .1049 .0509 .0530 .0404 .0775 Serv .0878 .1708 .0765 .0734 .1546 .1676 .1382 Oth .0001 .0054 .0008 .0055 .0183 .0250 .0012 TABLE 4 External Demands for 1967 U.S. Economy External Demands Agriculture 9300 Mining 1454 Construction 85,583 Manufacturing 278,797 Trade & transportation 141,468 Services 211,937 Others 2453 *Derived from more detailed tables published by the U.S. Bureau of Economic Analysis. Source for Tables 3 and 4: Input-Output Analysis: Foundations and Extensions, Ronald E. Miller and Peter D. Blair (Englewood Cliffs, N.J.: Prentice Hall, 1985)
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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
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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 and 96: Appendix C C-95 him/herself through
Page 97: Appendix C C-97 PROJECT The Leontie
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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