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Chapter 03 Matrices a b c d e ANSWER d 6 Write the transpose of matrix F a b c d Copyright Cengage Learning Powered by Cognero Page 3 ( Name ) ( Clas s ) ( Dat e ) ( ) ( Chapter 03 Matrices ) e ANSWER.
Nam e : Clas s: Dat e: Chapter 03: Matrices a b c d e d ANSWER: Write the transpose of matrix F a b c d Copyright Cengage Learning Powered by Cognero Page e c ANSWER: What is the sum of matrix M and its negative? a b c d e c ANSWER: If matrix M has element , what is ? a b c d e Copyright Cengage Learning Powered by Cognero Page a ANSWER: Use the following matrices to perform the indicated matrix operations, if possible ; a b c d e e ANSWER: 10 Use the following matrices to perform the indicated matrix operations, if possible ; a b Copyright Cengage Learning Powered by Cognero Page c d e e ANSWER: 11 Use the following matrices to perform the indicated matrix operations, if possible ; a b c d e ANSWER: b 12 Use the following matrices to perform the indicated matrix operations, if possible Copyright Cengage Learning Powered by Cognero Page ; a b c d e a ANSWER: 13 Use the following matrices to perform the indicated matrix operations, if possible a b c Copyright Cengage Learning Powered by Cognero Page d e c ANSWER: 14 Use the following matrices to perform the indicated matrix operations, if possible ; a b c d e ANSWER: d 15 Use the following matrices to perform the indicated matrix operations, if possible Copyright Cengage Learning Powered by Cognero Page ; a b c d is undefined e e ANSWER: 16 Use the following matrices to perform the indicated matrix operations, if possible a b c d e Copyright Cengage Learning Powered by Cognero Page a ANSWER: 17 Use the following matrices to perform the indicated matrix operations, if possible ; a b c d e b ANSWER: 18 Use the following matrices to perform the indicated matrix operations, if possible ; a b c Copyright Cengage Learning Powered by Cognero Page 10 d e c ANSWER: 19 Use the following matrices to perform the indicated matrix operations, if possible ; a b c d e c ANSWER: 20 Find x, y, z and w a , and Copyright Cengage Learning Powered by Cognero Page 11 b , and c , and d , and e , and d ANSWER: 21 Find x, y, z and w a and b and c and d and e and d ANSWER: 22 Solve for x, y and z a b and and c and d and e and a ANSWER: 23 Find x, y, z, and w a Copyright Cengage Learning Powered by Cognero and Page 12 201 The following technology matrix describes the relationship of certain industries within the economy to each other (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) Which is most dependent on the service industry? manufacturing utilities service fuels agriculture and food a b c d e c ANSWER: 202 The following technology matrix describes the relationship of certain industries within the economy to each other (A&F, agriculture and food; RM, raw materials; M, manufacturing; F, fuels industry; U, utilities; SI, service industries) Which industry would be most affected by a rise in the cost of utilites? a b c d e manufacturing utilities fuels service raw materials ANSWER: Copyright Cengage Learning Powered by Cognero d Page 114 203 Suppose a primitive economy consists of two industries, farm products and farm machinery Suppose also that its technology matrix is If surpluses of 97 units of farm products and 18 units of farm machinery are desired, find the gross production of each industry a b c d e ANSWER: 205 units of farm product, 60 units of machinery 260 units of farm product, 60 units of machinery 210 units of farm product, 55 units of machinery 205 units of farm product, 55 units of machinery 260 units of farm product, 270 units of machinery d 204 Suppose an economy has two industries, agriculture and minerals Suppose further that the technology matrix for this economy is A If surpluses of 40 agricultural units and 70 mineral units are desired, find the gross production to the nearest unit for each industry a b c d e ANSWER: agriculture = 167, minerals = 103 agriculture = 65, minerals = 167 agriculture = 71, minerals = 96 agriculture = 71, minerals = 103 agriculture = 65, minerals = 96 c 205 A primitive economy has a mining industry and a fishing industry, with technology matrix A If surpluses of 132 units of mining output and 27 units of fishing output are desired, find the gross production of Copyright Cengage Learning Powered by Cognero Page 115 each industry a b c d e mining = 180, fishing = 60 mining = 180, fishing = 240 mining = 200, fishing = 60 mining = 200, fishing = 70 mining = 240, fishing = 70 a ANSWER: 206 Suppose the economy of an underdeveloped country has an agricultural industry and an oil industry, with technology matrix A If surpluses of units of agricultural products and 610 units of oil products are desired, find the gross production of each industry to the nearest unit a b c d e agricultural = 800, oil = 825 agricultural = 110, oil = 700 agricultural = 110, oil = 715 agricultural = 100, oil = 715 agricultural = 100, oil = 700 e ANSWER: 207 Suppose a simple economy with only an agricultural industry and a steel industry has the following technology matrix If surpluses of 15 units of agricultural products and 34 units of steel are desired, find the gross production to the nearest unit of each industry a b c d e agriculture = 74, steel = 103 agriculture = 175, steel = 174 agriculture = 174, steel = 175 agriculture = 71, steel = 103 agriculture = 74, steel = 101 Copyright Cengage Learning Powered by Cognero Page 116 d ANSWER: 208 One sector of an economy consists of a mining industry and a manufacturing industry, with technology matrix A Surpluses of 32 units of mining output and 263 units of manufacturing output are desired Find the gross production of each industry a b c d e ANSWER: mining = 790, manufacturing = 500 mining = 290, manufacturing = 500 mining = 285, manufacturing = 790 mining = 790, manufacturing = 495 mining = 290, manufacturing = 495 b 209 An underdeveloped country has an agricultural industry and a manufacturing industry, with technology matrix A Surpluses of 10 units of agricultural products and 609 units of manufactured products are desired Find the gross production of each industry a b c d e ANSWER: agriculture = 160, manufacturing = 1210 agriculture = 1340, manufacturing = 1210 agriculture = 160, manufacturing = 1180 agriculture = 1340, manufacturing = 1180 agriculture = 170, manufacturing = 1210 c 210 A simple economy has an electronic components industry and a computers industry Each unit of electronic components output requires inputs of 0.3 unit of electronic components and 0.2 unit of computers Each unit of computer industry output requires inputs of 0.8 unit of electronic components and 0.1 unit of computers Write the technology matrix for this simple economy (E, electronics components industry; C, computer industry) Copyright Cengage Learning Powered by Cognero Page 117 a b c d e c ANSWER: 211 A simple economy has an electronic components industry and a computers industry Each unit of electronic components output requires inputs of 0.3 unit of electronic components and 0.6 unit of computers Each unit of computer industry output requires inputs of 0.2 unit of electronic components and 0.2 unit of computers If surpluses of 641 units of electronic components and 10 units of computers are desired, find the gross production to the nearest unit of each industry a b c d e ANSWER: electronics = 1170, computers = 865 electronics = 1170, computers = 890 electronics = 1207, computers = 2072 electronics = 1207, computers = 865 electronics = 2072, computers = 2060 b 212 An economy has an agricultural industry and a textile industry Each unit of agricultural output requires 0.4 unit of agricultural input and 0.2 unit of textiles input Each unit of textiles output requires 0.1 unit of agricultural input and 0.2 unit of textiles input Write the technology matrix for this economy (A, agricultural industry; T, textile industry) a Copyright Cengage Learning Powered by Cognero Page 118 b c d e e ANSWER: 213 An economy has an agricultural industry and a textile industry Each unit of agricultural output requires 0.4 unit of agricultural input and 0.1 unit of textiles input Each unit of textiles output requires 0.1 unit of agricultural input and 0.2 unit of textiles input If surpluses of units of agricultural products and 195 units of textiles are desired, find the gross production of each industry a b c d e agriculture = 60, textiles = 240 agriculture = 50, textiles = 250 agriculture = 60, textiles = 250 agriculture = 300, textiles = 250 agriculture = 60, textiles = 300 b ANSWER: 214 An economy has a manufacturing industry and a banking industry Each unit of manufacturing output requires inputs of 0.4 unit of manufacturing and 0.1 unit of banking Each unit of banking output requires inputs of 0.3 unit each of manufacturing and banking Write the technology matrix for this economy (M, manufacturing industry; B, banking industry) a b Copyright Cengage Learning Powered by Cognero Page 119 c d e d ANSWER: 215 An economy has a manufacturing industry and a banking industry Each unit of manufacturing output requires inputs of 0.5 unit of manufacturing and 0.2 unit of banking Each unit of banking output requires inputs of 0.3 unit each of manufacturing and banking If surpluses of 133 units of manufacturing and 100 units of banking are desired, find the gross production of each industry a b c d e ANSWER: manufacturing =424, banking =264 manufacturing = 390, banking = 320 manufacturing = 380, banking = 310 manufacturing = 380, banking = 320 manufacturing = 390, banking = 310 a 216 Suppose the development department of a firm charges 10% of its total monthly costs to the promotional department, and the promotional department charges 5% of its total monthly costs to the development department The direct costs of the development department are $25,400, and the direct costs of the promotional department are $9,400 The solution to the matrix equation or the column matrix gives the total costs for each department Note that this equation can be written in the form or , which is the form for an open economy Find the total costs for each department in this (micro)economy a b c d e ANSWER: development = 27,000, promotional = 11,000 development = 27,000, promotional = 12,000 development = 26,000, promotional = 12,000 development = 38,000, promotional = 12,000 development = 26,000, promotional = 11,000 Copyright Cengage Learning Powered by Cognero c Page 120 217 Suppose the shipping department of a firm charges 20% of its total monthly costs to the printing department and that the printing department charges 10% of its total monthly costs to the shipping department If the direct costs of the shipping department are $18,000 and the direct costs of the printing department are $11,000, find the total costs for each department Round your answer to the nearest hundredths a b c d e ANSWER: shipping = 21,500, printing = 14,000 shipping = 19,500, printing = 16,000 shipping = 21,500, printing = 15,000 shipping = 14,000, printing = 21,500 shipping = 19,500, printing = 15,000 e 218 The sales department of an auto dealership charges 10% of its total monthly costs to the service department, and the service department charges 20% of its total monthly costs to the sales department During a given month, the direct costs are $87,100 for sales and $49,600 for service Find the total costs of each department a b c d e sales = 69,500, service = 59,000 sales = 99,000, service = 59,500 sales = 87,100, service = 49,600 sales = 98,000, service = 59,000 sales = 99,000, service = 59,000 b ANSWER: 219 Suppose that an economy can be represented by the technology matrix below If surpluses of 180 units of fishing output, 93 units of agricultural goods, and 40 units of mining goods are desired, find the gross production to the nearest unit of each industry a b c d e ANSWER: fishing = 632, agriculture = 796, mining = 570 fishing = 633, agriculture = 796, mining = 568 fishing = 632, agriculture = 796, mining = 568 fishing = 632, agriculture = 793, mining = 568 fishing = 633, agriculture = 793, mining = 570 d 220 Suppose that the economy of a small nation has an electronics industry, a steel industry, and an auto Copyright Cengage Learning Powered by Cognero Page 121 industry, with the following technology matrix If the nation wishes to have surpluses of 104 units of electronics production, 271 units of steel production, and 200 automobiles, find the gross production of each industry a b c d e ANSWER: electronics = 1254, steel = 1262, autos = 722 electronics = 1252, steel = 1262, autos = 724 electronics = 1254, steel = 1269, autos = 722 electronics = 1254, steel = 1269, autos = 724 electronics = 1252, steel = 1262, autos = 722 e 221 Suppose an economy has the technology matrix below If surpluses of 525 units of electronics, 31 units of steel, and 140 autos are desired, find the gross production for each industry a b c d e ANSWER: electronics = 2150, steel = 985, autos = 690 electronics = 2150, steel = 2152, autos = 690 electronics = 2150, steel = 690, autos = 985 electronics = 2152, steel = 985, autos = 690 electronics = 2152, steel = 690, autos = 992 a 222 Suppose that an economy has the technology matrix below If surpluses of 56 units of manufactured goods and 104 units of agricultural goods are desired, find the gross production of each industry a service = 75, manufacturing = 220, agriculture = 200 Copyright Cengage Learning Powered by Cognero Page 122 service = 75, manufacturing = 220, agriculture = 210 c service = 70, manufacturing = 220, agriculture = 200 d service = 70, manufacturing = 250, agriculture = 200 e service = 75, manufacturing = 250, agriculture = 210 b c ANSWER: 223 Suppose the technology matrix for a closed model of a simple economy is given by Find the gross productions for the industries a b c d e products = households, machinery = products = households, machinery = households products = households, machinery = households products = households, machinery = products = households, machinery = ANSWER: households households households b 224 Suppose the technology matrix for a closed model of a simple economy is given by matrix A Find the gross productions for the industries Copyright Cengage Learning Powered by Cognero Page 123 a government = households, industry = households b government = households, industry = households c government = households, industry = households d government = households, industry = households e government = households, industry = households e ANSWER: 225 Suppose the technology matrix for a closed model of an economy is given by matrix A Find the gross productions for the industries a shipping = households, manufacturing = households shipping = households, manufacturing = households shipping = households, manufacturing = households b c Copyright Cengage Learning Powered by Cognero Page 124 d shipping = households, manufacturing = shipping = households, manufacturing = households e households d ANSWER: 226 A closed model for an economy has a manufacturing industry, utilities industry, and households industry Each unit of manufacturing output uses 0.6 unit of manufacturing input, 0.4 unit of utilities input, and 0.2 unit of households input Each unit of utilities output requires 0.3 unit of manufacturing input, 0.5 unit of utilities input, and 0.2 unit of households input Each unit of household output requires 0.2 unit each of manufacturing and utilities input and 0.4 unit of households input Write the augmented matrix for this closed model of the economy a b c d e ANSWER: b 227 A closed model for an economy has a manufacturing industry, utilities industry, and households industry Each unit of manufacturing output uses 0.5 unit of manufacturing input, 0.4 unit of utilities input, and 0.1 unit of households input Each unit of utilities output requires 0.4 unit of manufacturing input, 0.5 unit of utilities input, and 0.1 unit of households input Each unit of household output requires 0.4 unit each of manufacturing and utilities input and 0.2 unit of households input Find the gross production for each industry manufacturing = households, utilities = households b manufacturing = households, utilities = households a Copyright Cengage Learning Powered by Cognero Page 125 manufacturing = households, utilities = households d manufacturing = households, utilities = households e manufacturing = households, utilities = households c e ANSWER: 228 A closed model for an economy identifies government, the profit sector, the nonprofit sector, and households as its industries Each unit of government output requires 0.3 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input Each unit of profit sector output requires 0.2 unit of government input, 0.3 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.4 unit of households input Each unit of nonprofit sector output requires 0.2 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.6 unit of households input Each unit of households output requires 0.05 unit of government input, 0.2 unit of profit sector input, 0.2 unit of nonprofit sector input, and 0.25 unit of households input Write the augmented matrix for this closed model of the economy a b c d e ANSWER: c 229 A closed model for an economy identifies government, the profit sector, the nonprofit sector, and Copyright Cengage Learning Powered by Cognero Page 126 households as its industries Each unit of government output requires profit sector input, output requires and unit of nonprofit sector input, and unit of government input, unit of government input, unit of households input Each unit of profit sector unit of profit sector input, unit of nonprofit sector input, unit of households input Each unit of nonprofit sector output requires unit of profit sector input, of households output requires sector input, and unit of unit of nonprofit sector input, and unit of government input, unit of government input, unit of households input Each unit unit of profit sector input, unit of nonprofit unit of households input Find the gross production for each industry a government = households, profit = government = households, profit = households, nonprofit = households b households, nonprofit = households c government = households, profit = households, nonprofit = households households, nonprofit = households households, nonprofit = households d government = households, profit = e government = households, profit = ANSWER: a 230 Card tables are made by joining legs and a top using bolts The legs are each made from a steel rod The top has a frame made from steel rods A cover and four special clamps that brace the top and hold the legs are joined to the frame using a total of bolts The parts-listing matrix for the card table assembly is given by Copyright Cengage Learning Powered by Cognero Page 127 If an order is received for 10 card tables, legs, top, cover, clamps, and 12 bolts, how many of each primary assembly item are required to fill the order? a b c d e ANSWER: 88 rods, 12 covers, 51 clamps, 140 bolts 88 rods, 88 covers, 140 clamps, 140 bolts 51 rods, 140 covers, 51 clamps, 140 bolts 88 rods, 12 covers, 51 clamps, 140 bolts 140 rods, 12 covers, 88 clamps, 88 bolts d 231 A log carrier has a body made from a 4-ft length of reinforced cloth having a patch on each side and a dowel slid through each end to act as handles The parts-listing matrix for the log carrier is given by How many of each primary assembly items are required to fill an order for 500 log carriers and 30 handles? a b c d e ANSWER: 2060 handles, 970 cloths, 500 patches 1030 handles, 2060 cloths, 1000 patches 1030 handles, 500 cloths, 500 patches 1030 handles, 970 cloths, 2060 patches 1030 handles, 500 cloths, 2060 patches Copyright Cengage Learning Powered by Cognero c Page 128