Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)

The general solutions are \(\left\{ \begin{array}{l}{x_1} = - 4 + 2{x_2}\\{x_2}{\rm{ is free}}\\{x_3} = - 7\end{array} \right.\).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify pivot position

To identify the pivot and the pivot position, observe the leftmost column of the matrix (nonzero column), that is, the pivot column. At the top of this column, 1 is the pivot.

Step 2: Apply row operation

To obtain the pivot position, make the second term of the pivot column 0.

Use \({x_1}\) and \( - 2{x_2}\) terms in the first equation to eliminate \(3{x_1}\) and \( - 6{x_2}\) terms from the second equation. Add \( - 3\) times row 1 to row 2.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\0&0&1&{ - 7}\end{array}} \right]\)

Step 3: Apply row operation

To eliminate the \( - {x_3}\) term from the first equation, add rows 1 and 2.

\(\left( {\begin{array}{*{20}{c}}1&{ - 2}&0&{ - 4}\\0&0&1&{ - 7}\end{array}} \right)\)

Step 4: Mark the pivot positions in the matrix

Mark the nonzero leading entries in columns 1 and 3.

Step 5: Convert the matrix into the equation

To obtain the general solution of the system of equations, you must convert the augmented matrix into the system of equations.

Write the obtained matrix into the equation notation.

Step 6: Obtain the general solutions of the system of equations

According to the pivot positions in the obtained matrix, \({x_1}\) and \({x_3}\) are the basic variables. Also, \({x_2}\) is the free variable. By using the free variable, the basic variables can be solved.

Thus, the general solutions are shown below:

\(\left\{ \begin{array}{l}{x_1} = - 4 + 2{x_2}\\{x_2}{\rm{ is free}}\\{x_3} = - 7\end{array} \right.\)

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Linear Algebra and its Applications

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Short Answer

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)

Step by Step Solution

Step 1: Identify pivot position

Step 2: Apply row operation

Step 3: Apply row operation

Step 4: Mark the pivot positions in the matrix

Step 5: Convert the matrix into the equation

Step 6: Obtain the general solutions of the system of equations

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Linear Algebra and its Applications

Answers without the blur.

Short Answer

Find the general solutions of the systems whose augmented matrices are given in Exercises 10. 10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)

Step by Step Solution

Step 1: Identify pivot position

Step 2: Apply row operation

Step 3: Apply row operation

Step 4: Mark the pivot positions in the matrix

Step 5: Convert the matrix into the equation

Step 6: Obtain the general solutions of the system of equations

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Pure Maths

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)