Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 27E from Chapter 1.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 27E
Chapter:
Problem:
Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and dl Justify your answer.x1 + 3x2 = fcx1 + dx2 = g
Step-by-Step Solution
Given Information
We are given with following system of equations \[\begin{array}{l} {x_1} + 3{x_2} = f\\ c{x_1} + d{x_2} = g \end{array}\] We have to find information about the coefficients $c$ and $d$, if the system of equations is consistent \\
Step-1: The Augmented matrix
\[\left[ {\begin{array}{*{20}{c}} 1&3&f\\ c&d&g \end{array}} \right]\] The echelon form of the augmented matrix \[\left[ {\begin{array}{*{20}{c}} 1&3&f\\ 0&{d - 3c}&{g - cf} \end{array}} \right]::\,\,{R_2} = {R_2} - c{R_1}\]
Step-2: The Augmented matrix
Let the constant in the second row is nonzero. If the system has to be consistent, then if $d-3c=0$, then we have \[0 = g - cf\] The above is not possible, therefore, for an consistent solutions, the term $d - 3c \ne 0$
If $d \ne 3c$, the system is consistent for all values of f and g
We are given with following system of equations \[\begin{array}{l} {x_1} + 3{x_2} = f\\ c{x_1} + d{x_2} = g \end{array}\] We have to find information about the coefficients $c$ and $d$, if the system of equations is consistent \\
Step-1: The Augmented matrix
\[\left[ {\begin{array}{*{20}{c}} 1&3&f\\ c&d&g \end{array}} \right]\] The echelon form of the augmented matrix \[\left[ {\begin{array}{*{20}{c}} 1&3&f\\ 0&{d - 3c}&{g - cf} \end{array}} \right]::\,\,{R_2} = {R_2} - c{R_1}\]
Step-2: The Augmented matrix
Let the constant in the second row is nonzero. If the system has to be consistent, then if $d-3c=0$, then we have \[0 = g - cf\] The above is not possible, therefore, for an consistent solutions, the term $d - 3c \ne 0$
If $d \ne 3c$, the system is consistent for all values of f and g