In Exercises 5-6 an elementary matrix E and a matrix A are given. Identify the row operation corresponding to E and verify that the product EA results from applying the row operation to A. 5. a. E = 0 [ J }]}, 1 0 1 0-3 b. E= 0 1 c. E = 0 0 4 1 0 00 1 A = 0 0 1 = [ " -1 -2 5 3 -6 -6 2 A1 2 1 4 A = 2 5 3 6 =] -1 0 -4 -4 -3 -1 5 3 013 -1
In Exercises 5-6 an elementary matrix E and a matrix A are given. Identify the row operation corresponding to E and verify that the product EA results from applying the row operation to A. 5. a. E = 0 [ J }]}, 1 0 1 0-3 b. E= 0 1 c. E = 0 0 4 1 0 00 1 A = 0 0 1 = [ " -1 -2 5 3 -6 -6 2 A1 2 1 4 A = 2 5 3 6 =] -1 0 -4 -4 -3 -1 5 3 013 -1
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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![**Elementary Matrices and Row Operations: Example Exercises**
In Exercises 5-6, an elementary matrix \( E \) and a matrix \( A \) are given. Identify the row operation corresponding to \( E \) and verify that the product \( EA \) results from applying the row operation to \( A \).
### 5.
**a.**
\[ E = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}, \quad A = \begin{bmatrix}
-1 & -2 & 5 & -1 \\
3 & -6 & -6 & -6
\end{bmatrix} \]
**b.**
\[ E = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -3 & 1
\end{bmatrix}, \quad A = \begin{bmatrix}
2 & -1 & 0 & -4 & -4 \\
1 & -3 & -1 & 5 & 3 \\
2 & 0 & 1 & 3 & -1
\end{bmatrix} \]
**c.**
\[ E = \begin{bmatrix}
1 & 0 & 4 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}, \quad A = \begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix} \]
### Explanation:
In these exercises, we are given an elementary matrix \( E \) and a matrix \( A \). The task is to identify the row operation represented by \( E \) and confirm that the product \( EA \) correctly represents the application of this row operation to the matrix \( A \).
#### Elementary Matrix Operations:
1. **Row Interchange (a):**
The matrix \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) swaps the first and second rows of matrix \( A \).
2. **Row Scaling and Addition (b):**
The matrix \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1 \end{bmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2F7ca6b59b-2ca0-4ed6-851b-ce57cea82084%2Fwhbebq_processed.png&w=3840&q=75)
Transcribed Image Text:**Elementary Matrices and Row Operations: Example Exercises**
In Exercises 5-6, an elementary matrix \( E \) and a matrix \( A \) are given. Identify the row operation corresponding to \( E \) and verify that the product \( EA \) results from applying the row operation to \( A \).
### 5.
**a.**
\[ E = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}, \quad A = \begin{bmatrix}
-1 & -2 & 5 & -1 \\
3 & -6 & -6 & -6
\end{bmatrix} \]
**b.**
\[ E = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -3 & 1
\end{bmatrix}, \quad A = \begin{bmatrix}
2 & -1 & 0 & -4 & -4 \\
1 & -3 & -1 & 5 & 3 \\
2 & 0 & 1 & 3 & -1
\end{bmatrix} \]
**c.**
\[ E = \begin{bmatrix}
1 & 0 & 4 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}, \quad A = \begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix} \]
### Explanation:
In these exercises, we are given an elementary matrix \( E \) and a matrix \( A \). The task is to identify the row operation represented by \( E \) and confirm that the product \( EA \) correctly represents the application of this row operation to the matrix \( A \).
#### Elementary Matrix Operations:
1. **Row Interchange (a):**
The matrix \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) swaps the first and second rows of matrix \( A \).
2. **Row Scaling and Addition (b):**
The matrix \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1 \end{bmatrix}
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