3. For each matrix shown, choose one of the following phrases that best describes the matrix. Give a brief justification for your choice. (a) The matrix is not in echelon form. (b) The matrix is in echelon form, but not in reduced echelon form. (c) The matrix is in reduced echelon form. 1 1 0 0 2 1 0 1 0 3 2 0 A = 0 B = 1 0 0 0 0 1 0 1 0 0 0 0 000 0 0 0 0 1 1 0 01 0 0 C = D=
3. For each matrix shown, choose one of the following phrases that best describes the matrix. Give a brief justification for your choice. (a) The matrix is not in echelon form. (b) The matrix is in echelon form, but not in reduced echelon form. (c) The matrix is in reduced echelon form. 1 1 0 0 2 1 0 1 0 3 2 0 A = 0 B = 1 0 0 0 0 1 0 1 0 0 0 0 000 0 0 0 0 1 1 0 01 0 0 C = D=
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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![### Matrices and Echelon Forms
**Question:**
For each matrix shown, choose one of the following phrases that best describes the matrix. Give a brief justification for your choice.
(a) The matrix is not in echelon form.
(b) The matrix is in echelon form, but not in reduced echelon form.
(c) The matrix is in reduced echelon form.
**Matrices:**
\[ A = \begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} \]
\[ B = \begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} \]
\[ C = \begin{bmatrix}
2 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix} \]
\[ D = \begin{bmatrix}
1 & 0 & 3 & 2 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix} \]
**Explanations:**
1. **Matrix A:**
- *Answer:* (b) The matrix is in echelon form, but not in reduced echelon form.
- *Justification:* All non-zero rows are above zero rows, and the leading entry in each non-zero row is to the right of the leading entry in the previous row. However, it is not in reduced echelon form because the leading coefficient in the first row is not 1.
2. **Matrix B:**
- *Answer:* (a) The matrix is not in echelon form.
- *Justification:* The first row is entirely zero, which violates the echelon form condition that every non-zero row should be above any rows of all zeros.
3. **Matrix C:**
- *Answer:* (a) The matrix is not in echelon form.
- *Justification:* The second row contains a leading 1 that is not to the right of the leading entry in the first row—the entry is missing.
4. **Matrix D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2F7a946e6e-c610-4a94-a315-b71274b69244%2F5blii5rs_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrices and Echelon Forms
**Question:**
For each matrix shown, choose one of the following phrases that best describes the matrix. Give a brief justification for your choice.
(a) The matrix is not in echelon form.
(b) The matrix is in echelon form, but not in reduced echelon form.
(c) The matrix is in reduced echelon form.
**Matrices:**
\[ A = \begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} \]
\[ B = \begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} \]
\[ C = \begin{bmatrix}
2 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix} \]
\[ D = \begin{bmatrix}
1 & 0 & 3 & 2 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix} \]
**Explanations:**
1. **Matrix A:**
- *Answer:* (b) The matrix is in echelon form, but not in reduced echelon form.
- *Justification:* All non-zero rows are above zero rows, and the leading entry in each non-zero row is to the right of the leading entry in the previous row. However, it is not in reduced echelon form because the leading coefficient in the first row is not 1.
2. **Matrix B:**
- *Answer:* (a) The matrix is not in echelon form.
- *Justification:* The first row is entirely zero, which violates the echelon form condition that every non-zero row should be above any rows of all zeros.
3. **Matrix C:**
- *Answer:* (a) The matrix is not in echelon form.
- *Justification:* The second row contains a leading 1 that is not to the right of the leading entry in the first row—the entry is missing.
4. **Matrix D
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