Find the general solution of the system whose augmented matrix is given below. 1 -1 2 -1 -17 0 5 5 12 -3 -3 1-2 -2 -2 -5 3 2 0 -6 -
Find the general solution of the system whose augmented matrix is given below. 1 -1 2 -1 -17 0 5 5 12 -3 -3 1-2 -2 -2 -5 3 2 0 -6 -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Solving Systems of Linear Equations Using Augmented Matrices
In linear algebra, systems of linear equations can be represented using augmented matrices. Below, we have an example of a system of linear equations represented by its augmented matrix.
### Problem Statement
Find the general solution of the system whose augmented matrix is given below:
\[
\begin{bmatrix}
1 & -1 & 2 & -1 & -1 \\
-5 & 0 & 0 & 5 & 5 \\
3 & -6 & 12 & -3 & -3 \\
2 & 1 & -2 & -2 & -2
\end{bmatrix}
\]
### Explanation
The given matrix represents a system of linear equations where the last column is the augmented part (representing the constants after the equals sign in each equation). This final column demonstrates how to convert the matrix into an augmented representation of the equations.
The matrix can be interpreted as follows:
1. \( 1x_1 - 1x_2 + 2x_3 - 1x_4 = -1 \)
2. \( -5x_1 = 5 \)
3. \( 3x_1 - 6x_2 + 12x_3 - 3x_4 = -3 \)
4. \( 2x_1 + 1x_2 - 2x_3 - 2x_4 = -2 \)
### Steps to Solve the Augmented Matrix
To find the solution to this system, typically, one would perform row operations to reduce the augmented matrix to its reduced row echelon form (RREF). These steps include:
1. Swapping rows.
2. Multiplying or dividing rows by non-zero constants.
3. Adding or subtracting multiples of rows from other rows.
Once in RREF, the matrix will indicate the solution to the system, either as unique, infinite, or no solution.
### Tips for Solving Augmented Matrices
1. Identify pivot positions and aim to create zeros above and below each pivot.
2. Use elementary row operations systematically.
3. Verify each row operation to avoid arithmetic errors.
4. Once in RREF, read off the solutions directly from the matrix.
Understanding how to transform augmented matrices and interpret them is crucial for solving systems of linear equations efficiently. By practicing these methods, you can develop a strong foundation in linear](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc7f96b0-6dc0-4972-bfe0-9c219792bf84%2F2945b769-d56e-435a-998d-02c05cafbd53%2Fb83z4as_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving Systems of Linear Equations Using Augmented Matrices
In linear algebra, systems of linear equations can be represented using augmented matrices. Below, we have an example of a system of linear equations represented by its augmented matrix.
### Problem Statement
Find the general solution of the system whose augmented matrix is given below:
\[
\begin{bmatrix}
1 & -1 & 2 & -1 & -1 \\
-5 & 0 & 0 & 5 & 5 \\
3 & -6 & 12 & -3 & -3 \\
2 & 1 & -2 & -2 & -2
\end{bmatrix}
\]
### Explanation
The given matrix represents a system of linear equations where the last column is the augmented part (representing the constants after the equals sign in each equation). This final column demonstrates how to convert the matrix into an augmented representation of the equations.
The matrix can be interpreted as follows:
1. \( 1x_1 - 1x_2 + 2x_3 - 1x_4 = -1 \)
2. \( -5x_1 = 5 \)
3. \( 3x_1 - 6x_2 + 12x_3 - 3x_4 = -3 \)
4. \( 2x_1 + 1x_2 - 2x_3 - 2x_4 = -2 \)
### Steps to Solve the Augmented Matrix
To find the solution to this system, typically, one would perform row operations to reduce the augmented matrix to its reduced row echelon form (RREF). These steps include:
1. Swapping rows.
2. Multiplying or dividing rows by non-zero constants.
3. Adding or subtracting multiples of rows from other rows.
Once in RREF, the matrix will indicate the solution to the system, either as unique, infinite, or no solution.
### Tips for Solving Augmented Matrices
1. Identify pivot positions and aim to create zeros above and below each pivot.
2. Use elementary row operations systematically.
3. Verify each row operation to avoid arithmetic errors.
4. Once in RREF, read off the solutions directly from the matrix.
Understanding how to transform augmented matrices and interpret them is crucial for solving systems of linear equations efficiently. By practicing these methods, you can develop a strong foundation in linear
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