Find the general solutions of the systems whose augmented matrices are given. To -4 2 1 352 2 -4 A = B = -8:30 C= 2 -3 2 2 305 1 2 -2 4 2 -1 ܥ ~3~
Find the general solutions of the systems whose augmented matrices are given. To -4 2 1 352 2 -4 A = B = -8:30 C= 2 -3 2 2 305 1 2 -2 4 2 -1 ܥ ~3~
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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![### Problem 5: Finding General Solutions of Systems with Given Augmented Matrices
The goal of this problem is to find the general solutions of the systems whose augmented matrices are provided below.
#### Augmented Matrices:
1. Matrix \( A \):
\[
A = \begin{bmatrix}
1 & 3 & 5 & 2 \\
2 & 3 & 0 & 5
\end{bmatrix}
\]
2. Matrix \( B \):
\[
B = \begin{bmatrix}
2 & 3 & -4 & 7 \\
1 & 2 & -2 & 4
\end{bmatrix}
\]
3. Matrix \( C \):
\[
C = \begin{bmatrix}
0 & -4 & 2 & 2 \\
2 & -3 & 2 & 3 \\
1 & 2 & -1 & -2
\end{bmatrix}
\]
### Instructions:
To find the general solutions for each system represented by the given augmented matrices, one usually performs row reduction to bring the matrix into a row echelon form or reduced row echelon form. From there, the general solution can be derived. Here’s a brief outline of the steps to be followed:
1. **Row Reduction**: Apply elementary row operations to convert the augmented matrix to its reduced row echelon form (RREF).
2. **Interpretation**: Once in RREF, interpret the results to find the solutions of the corresponding linear system.
3. **General Solution**: Express the general solution in parametric form if the system has infinitely many solutions.
#### Detailed Interpretation of Graphs and/or Diagrams:
- **Matrix \( A \)** and **Matrix \( B \)**: These matrices are 2x4 augmented matrices where the first three columns represent the coefficients of the variables, and the last column represents the constants from the corresponding systems of equations.
- **Matrix \( C \)**: This is a 3x4 augmented matrix that includes three equations with three variables and constants, thus a larger system compared to \( A \) and \( B \).
By working through these steps for each matrix, you will identify whether each system is consistent (has at least one solution) or inconsistent (has no solution), and then describe the solutions accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2F5a491237-be89-48ff-85bd-aa8ae38d551b%2Ftw9jl49_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 5: Finding General Solutions of Systems with Given Augmented Matrices
The goal of this problem is to find the general solutions of the systems whose augmented matrices are provided below.
#### Augmented Matrices:
1. Matrix \( A \):
\[
A = \begin{bmatrix}
1 & 3 & 5 & 2 \\
2 & 3 & 0 & 5
\end{bmatrix}
\]
2. Matrix \( B \):
\[
B = \begin{bmatrix}
2 & 3 & -4 & 7 \\
1 & 2 & -2 & 4
\end{bmatrix}
\]
3. Matrix \( C \):
\[
C = \begin{bmatrix}
0 & -4 & 2 & 2 \\
2 & -3 & 2 & 3 \\
1 & 2 & -1 & -2
\end{bmatrix}
\]
### Instructions:
To find the general solutions for each system represented by the given augmented matrices, one usually performs row reduction to bring the matrix into a row echelon form or reduced row echelon form. From there, the general solution can be derived. Here’s a brief outline of the steps to be followed:
1. **Row Reduction**: Apply elementary row operations to convert the augmented matrix to its reduced row echelon form (RREF).
2. **Interpretation**: Once in RREF, interpret the results to find the solutions of the corresponding linear system.
3. **General Solution**: Express the general solution in parametric form if the system has infinitely many solutions.
#### Detailed Interpretation of Graphs and/or Diagrams:
- **Matrix \( A \)** and **Matrix \( B \)**: These matrices are 2x4 augmented matrices where the first three columns represent the coefficients of the variables, and the last column represents the constants from the corresponding systems of equations.
- **Matrix \( C \)**: This is a 3x4 augmented matrix that includes three equations with three variables and constants, thus a larger system compared to \( A \) and \( B \).
By working through these steps for each matrix, you will identify whether each system is consistent (has at least one solution) or inconsistent (has no solution), and then describe the solutions accordingly.
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