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 19E from Chapter 1.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 19E
Chapter:
Problem:
In Exercises 19-22, determine the value(s) of hsuch that the matrix is the augmented matrix of a consistent linear system...
Step-by-Step Solution
Step 1
Given Augmented Matrix:
\[\left[ \begin{array} { l l l } { 1 } & { h } & { 4 } \\ { 3 } & { 6 } & { 8 } \end{array} \right]\]We have to determine the value of h for which the given system of equations is consistent. To be a consistent system, the system should have a solution.
Step 2: Echelon form of the matrix
\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{l}}1&h&4\\3&6&8\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{l}}1&h&4\\0&{6 - 3h}&{ - 4}\end{array}} \right]::\,\,{R_3} = {R_3} - 3{R_1}\end{array}\]
Step 3: Required value of $h$
The system may have an inconsistent solution if in the last row, the second element is zero. So to make the system consistent, the second element in last row should be non zero. \[\begin{array}{l}6 - 3h \ne 0\\\\3h \ne 6\\\\h \ne 2\end{array}\]
ANSWERS
Hence the system is consistent except when h=2. \[h \ne 2\]
Given Augmented Matrix:
\[\left[ \begin{array} { l l l } { 1 } & { h } & { 4 } \\ { 3 } & { 6 } & { 8 } \end{array} \right]\]We have to determine the value of h for which the given system of equations is consistent. To be a consistent system, the system should have a solution.
Step 2: Echelon form of the matrix
\[\begin{array}{l}A = \left[ {\begin{array}{*{20}{l}}1&h&4\\3&6&8\end{array}} \right]\\\\ = \left[ {\begin{array}{*{20}{l}}1&h&4\\0&{6 - 3h}&{ - 4}\end{array}} \right]::\,\,{R_3} = {R_3} - 3{R_1}\end{array}\]
Step 3: Required value of $h$
The system may have an inconsistent solution if in the last row, the second element is zero. So to make the system consistent, the second element in last row should be non zero. \[\begin{array}{l}6 - 3h \ne 0\\\\3h \ne 6\\\\h \ne 2\end{array}\]
ANSWERS
Hence the system is consistent except when h=2. \[h \ne 2\]