Explanation Strategy for finding the inverse of a matrix: If a sequence of a row operations performed on the n × n matrix A reduces A to the n × n identity matrix I , then those same row operations, performed in the same order on I , will transform I into A − 1 . If no sequence of row operations will reduce A to I , then A is not invertible. A notation for this process uses an n -row-by- 2 n -column matrix, with matrix A as the left half and matrix I as the right half. If A is invertible, the proper row operations on [ A | I ] will transfer it into [ I | A − 1 ] . Given: The numbers of hours required to manufacture and test each of two models of heart monitor are given in the first table and the number of hours available each week for manufacturing and testing in the second table. Hours required per unit: Model A Model B Manufacturing 23 27 Testing 21 22 Time Available: Manufacturing 127 Testing 108 Calculation: Let x , y , and z represent the number of units A and B to be manufactured respectively. We can set up the following system of equations: 23 hours is needed for every A and 27 hours is needed for every B. 23 x + 27 y = 127 21 hours is needed for every A and 22 hours is needed for every B. 21 x + 22 y = 108 A system of equation can always be written as a matrix equation A X = B , where A is the coefficient matrix of the system, X is a column matrix of variables, and B is the column matrix of constants. In matrix form, the system becomes, [ 23 27 21 22 ] [ x y ] = [ 127 108 ] If matrix A is invertible, the matrix equation A X = B is easy to solve, as we see in the next step. First we find the inverse of matrix A . Let A = [ 23 27 21 22 ] We can set up a 2 × 4 matrix with A on the left and I on the right of the line, [ A | I ] = [ 23 27 21 22 | 1 0 0 1 ] We perform row operations on the entire matrix to transform the left half into I . [ A | I ] = [ 23 27 21 22 | 1 0 0 1 ] R 1 → 1 23 ⋅ R 1 [ 1 27 23 21 22 | 1 23 0 0 1 ] R 2 → R 2 − 21 ⋅ R 1 [ 1 27 23 0 − 61 23 | 1 23 0 − 21 23 1 ]

Bundle: Webassign Printed Access Card For Gustafson/hughes' College Algebra, Single-term + Student Solutions Manual For Gustafson/hughes? College Algebra, 12th

12th Edition

ISBN: 9781337605304

Author: R. David Gustafson; Jeff Hughes

Publisher: Cengage Learning

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Chapter 6.4, Problem 37E

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How many of each model can be manufactured each week.

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Chapter 6.4, Problem 36E

Chapter 6.4, Problem 38E

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Chapter 6 Solutions

Bundle: Webassign Printed Access Card For Gustafson/hughes' College Algebra, Single-term + Student Solutions Manual For Gustafson/hughes? College Algebra, 12th

Ch. 6.1 - Self Check Use the graphing method to solve...Ch. 6.1 - Prob. 2SC Ch. 6.1 - Prob. 3SC Ch. 6.1 - Prob. 4SC Ch. 6.1 - Self Check 5 Use the elimination method to solve...Ch. 6.1 - Prob. 6SC Ch. 6.1 - Prob. 7SC Ch. 6.1 - Prob. 8SC Ch. 6.1 - Fill in the blanks. Any set of several equations...Ch. 6.1 - Prob. 2E

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Solution Summary: The author explains how to find the inverse of a matrix by calculating the number of hours required to manufacture and test each of two models of heart monitor.

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Chapter 6 Solutions

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How many of each model can be manufactured each week.