Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. Find the echelon form of the given matrix. 1 4 -2 3 3-11 32 95 25 O1 4-2 37 0134 0-69-7 O1 4-2 37 0134 0 0 15-1 O1 4-2 37 01 34 0 0 27 0 O14-2.3] 0134 0 0 27 17
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. Find the echelon form of the given matrix. 1 4 -2 3 3-11 32 95 25 O1 4-2 37 0134 0-69-7 O1 4-2 37 0134 0 0 15-1 O1 4-2 37 01 34 0 0 27 0 O14-2.3] 0134 0 0 27 17
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Quiz Instructions
This quiz covers sections 1.1 and 1.2.
### Question 7
**Problem Statement:**
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
Find the echelon form of the given matrix.
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
-3 & -11 & 9 & -5 \\
2 & 2 & 5 & -1 \\
\end{bmatrix}
\]
**Options:**
- \(\text{Option A:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & -6 & 9 & -7 \\
\end{bmatrix}
\]
- \(\text{Option B:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 15 & -1 & 0 \\
\end{bmatrix}
\]
- \(\text{Option C:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 27 & 0 & 0 \\
\end{bmatrix}
\]
- \(\text{Option D:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 27 & 17 & 0 \\
\end{bmatrix}
\]
**Instructions:**
Select the option that correctly represents the echelon form of the given matrix.
**Explanation of Diagrams:**
The problem does not contain any graphs or diagrams. The matrix and the answer options are presented in a concise mathematical format. The numbers within the matrices represent the elements that need to be transformed according to the row reduction algorithm to reach echelon form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2Ffd82fc5f-3729-4629-9d5b-614200da300e%2F0temhrn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quiz Instructions
This quiz covers sections 1.1 and 1.2.
### Question 7
**Problem Statement:**
Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.
Find the echelon form of the given matrix.
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
-3 & -11 & 9 & -5 \\
2 & 2 & 5 & -1 \\
\end{bmatrix}
\]
**Options:**
- \(\text{Option A:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & -6 & 9 & -7 \\
\end{bmatrix}
\]
- \(\text{Option B:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 15 & -1 & 0 \\
\end{bmatrix}
\]
- \(\text{Option C:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 27 & 0 & 0 \\
\end{bmatrix}
\]
- \(\text{Option D:}\)
\[
\begin{bmatrix}
1 & 4 & -2 & 3 \\
0 & 1 & 3 & 4 \\
0 & 27 & 17 & 0 \\
\end{bmatrix}
\]
**Instructions:**
Select the option that correctly represents the echelon form of the given matrix.
**Explanation of Diagrams:**
The problem does not contain any graphs or diagrams. The matrix and the answer options are presented in a concise mathematical format. The numbers within the matrices represent the elements that need to be transformed according to the row reduction algorithm to reach echelon form.
![### Question 6
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
\[
\begin{bmatrix}
1 & 4 & 5 & -7 \\
0 & 1 & -4 & -6 \\
0 & 2 & 1 & 6
\end{bmatrix}
\]
- [ ] Reduced echelon form
- [ ] Neither
- [ ] Echelon form
#### Explanation of Matrix Forms
**Echelon Form:**
1. All zero rows, if any, are at the bottom of the matrix.
2. The leading entry of each nonzero row (also known as a pivot) is to the right of the leading entry of the row above it.
3. The leading entry in any nonzero row is 1.
**Reduced Echelon Form:**
In addition to the conditions above:
1. Each leading 1 is the only non-zero entry in its column.
**Analysis of the Given Matrix:**
- The leading entry in the first row is 1.
- The leading entry in the second row is 1, and it is positioned to the right of the leading entry in the first row.
- The leading entry in the third row is 2, which does not satisfy the condition that the leading entry should be 1.
By this analysis, it's evident that the matrix satisfies the conditions for Echelon form but not Reduced Echelon form, as not all leading entries are 1 where required and the columns containing leading entries are not solely non-zero in their position.
#### Answer:
- [ ] Reduced echelon form
- [ ] Neither
- [x] Echelon form](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2Ffd82fc5f-3729-4629-9d5b-614200da300e%2Ffczair_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 6
Determine whether the matrix is in echelon form, reduced echelon form, or neither.
\[
\begin{bmatrix}
1 & 4 & 5 & -7 \\
0 & 1 & -4 & -6 \\
0 & 2 & 1 & 6
\end{bmatrix}
\]
- [ ] Reduced echelon form
- [ ] Neither
- [ ] Echelon form
#### Explanation of Matrix Forms
**Echelon Form:**
1. All zero rows, if any, are at the bottom of the matrix.
2. The leading entry of each nonzero row (also known as a pivot) is to the right of the leading entry of the row above it.
3. The leading entry in any nonzero row is 1.
**Reduced Echelon Form:**
In addition to the conditions above:
1. Each leading 1 is the only non-zero entry in its column.
**Analysis of the Given Matrix:**
- The leading entry in the first row is 1.
- The leading entry in the second row is 1, and it is positioned to the right of the leading entry in the first row.
- The leading entry in the third row is 2, which does not satisfy the condition that the leading entry should be 1.
By this analysis, it's evident that the matrix satisfies the conditions for Echelon form but not Reduced Echelon form, as not all leading entries are 1 where required and the columns containing leading entries are not solely non-zero in their position.
#### Answer:
- [ ] Reduced echelon form
- [ ] Neither
- [x] Echelon form
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