VIPSolutionsGive step-by-step solution with explanation and final answer: Please do not include two equal sign in the same line.(2) Determine if the augmented matrices give below are in row echelon form (REF). If it is not in REF, explain why not. If it is in REF, determine if they are in reduced row-echelon form (RREF). If it is not in RREF, explain why not. Please note that in lieu of vertical line, — is used for conve- nience. (a) 1000 | 7 ; 11 2 | J 0000] 3 (b) 10 20 | 7 b 11 2 | | 0-100 | 3 (c) 10 00 | 7 0-1 12 | 1 00 003 (d) 2 1 0 | 5 0 -1 2 | 11 1 3 5] 0
Question:
Give step-by-step solution with explanation and final answer:
Please do not include two equal sign in the same line.
(2) Determine if the augmented matrices give below are in row echelon form
(REF). If it is not in REF, explain why not. If it is in REF, determine if
they are in reduced row-echelon form (RREF). If it is not in RREF, explain
why not.
Please note that in lieu of vertical line, — is used for conve-
nience.
(a)
1000 | 7
; 11 2 | J
0000] 3
(b)
10 20 | 7
b 11 2 | |
0-100 | 3
(c)
10 00 | 7
0-1 12 | 1
00 003
(d)
2 1 0 | 5
0 -1 2 | 11
1 3 5] 0
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Asked by: MD ISRARUL HAQUE
Created at: 2025-09-10 06:29:46
Asked by: MD ISRARUL HAQUE
Created at: 2025-09-10 06:29:46
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# Step-by-Step Solution
Let's recall the definitions first:
## Row Echelon Form (REF) conditions:
1. All nonzero rows are above any rows of all zeros.
2. The leading entry of each nonzero row after the first occurs to the right of the leading entry of the previous row.
3. All entries in a column below a leading entry are zero.
## Reduced Row Echelon Form (RREF) conditions:
- All conditions for REF, **and**:
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
---
## (a)
\[
\begin{bmatrix}
1 & 0 & 0 & 0 & 7 \\
0 & 1 & 1 & 2 & 1 \\
0 & 0 & 0 & 0 & 3
\end{bmatrix}
\]
### Check REF:
1. All nonzero rows are above any all-zero row (Yes: row 3 is not all zeros).
2. Leading entries: Row 1: 1, Row 2: 1 (column 2), Row 3: 3 (last column, not a variable, **so not considered leading entry for variables**). So, row 3 does not contain a leading entry for a variable, just an augmented value.
3. All entries below leading entries are zero (Yes).
**This matrix is in REF.**
### Check RREF:
- Row 2's leading 1 (column 2) has nonzero entry above (row 1, col 2 = 0).
- For row 2, leading 1 is in column 2, but columns 3 and 4 have nonzero entries in row 2.
- **Column 3 and 4's leading variable is not a 1, and the leading 1 in row 2 is not the only nonzero entry in its column.**
**Not in RREF.**
- **Reason**: The leading 1 in row 2 is not the only nonzero entry in its row (row 2).
---
## (b)
\[
\begin{bmatrix}
1 & 0 & 2 & 0 & 7 \\
0 & 1 & 1 & 2 & 1 \\
0 & -1 & 0 & 0 & 3
\end{bmatrix}
\]
### Check REF:
1. All nonzero rows above any all-zero row (Yes).
2. Leading entries: Row 1: 1 (col 1), Row 2: 1 (col 2), Row 3: -1 (col 2).
3. **Row 3's leading entry (-1) is in the same column as row 2's leading entry (not to the right).**
**Not in REF.**
- **Reason**: The leading entry in row 3 is not to the right of the leading entry in row 2.
---
## (c)
\[
\begin{bmatrix}
1 & 0 & 0 & 0 & 7 \\
0 & -1 & 1 & 2 & 1 \\
0 & 0 & 0 & 0 & 3
\end{bmatrix}
\]
### Check REF:
1. All nonzero rows above any all-zero row (Yes).
2. Leading entries: Row 1: 1 (col 1), Row 2: -1 (col 2). Row 3: just a number in augmented part.
3. All entries below leading entries are zero (Yes).
**This matrix is in REF.**
### Check RREF:
- Row 2's leading entry is -1, **not 1**.
**Not in RREF.**
- **Reason**: The leading entry in row 2 is not 1.
---
## (d)
\[
\begin{bmatrix}
-2 & 1 & 0 & 5 \\
0 & -1 & 2 & 11 \\
1 & 3 & 5 & 0
\end{bmatrix}
\]
### Check REF:
1. All nonzero rows above any all-zero row (Yes).
2. Leading entries: Row 1: -2 (col 1), Row 2: -1 (col 2), Row 3: 1 (col 1). **Row 3's leading entry is in the same column as row 1 (not to the right).**
**Not in REF.**
- **Reason**: The leading entry in row 3 is not to the right of the leading entry in row 2.
---
# **Final Answers (Summary Table)**
| Part | In REF? | Why/Why not? | In RREF? | Why/Why not? |
|------|-----------|----------------------------------------------------------------------|--------------|----------------------------------------------|
| (a) | Yes | All conditions for REF satisfied | No | Row 2's leading 1 not only nonzero in row |
| (b) | No | Row 3's leading entry not to right of row 2's leading entry | No | Not in REF |
| (c) | Yes | All conditions for REF satisfied | No | Row 2's leading entry is -1, not 1 |
| (d) | No | Row 3's leading entry not to right of previous leading entry | No | Not in REF |
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