Row Reduction (2)
From a General Matrix to the Reduced Echelon Form
The colors below mean the following:
Blue=Done Part, Yellow=Current Focus, Pink=Pivot
The Given Matrix
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
3 |
| 0 |
0 |
0 |
-3 |
-6 |
-3 |
-6 |
3 |
0 |
3 |
-6 |
-3 |
| 0 |
0 |
2 |
-1 |
3 |
3 |
4 |
-3 |
-1 |
0 |
-4 |
1 |
| 0 |
0 |
6 |
-11 |
-7 |
1 |
-4 |
-1 |
0 |
8 |
-4 |
1 |
| 0 |
0 |
4 |
-7 |
-4 |
-1 |
4 |
-3 |
-7 |
1 |
-6 |
3 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
Objective:
Obtain the reduced echelon form of the given
matrix by elementary row operations.
Three types of row operations are allowed:
- (Row i) replaced by (Row i)+c(Row j), where i≠j and c≠0.
- (Row i) replaced by c(Row i), where c≠0.
- Interchange two rows, or rearrange the order of several rows.
| 0 |
0 |
2 |
-1 |
3 |
3 |
4 |
-3 |
-1 |
0 |
-4 |
1 |
| 0 |
0 |
0 |
-3 |
-6 |
-3 |
-6 |
3 |
0 |
3 |
-6 |
-3 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
3 |
| 0 |
0 |
6 |
-11 |
-7 |
1 |
-4 |
-1 |
0 |
8 |
-4 |
1 |
| 0 |
0 |
4 |
-7 |
-4 |
-1 |
4 |
-3 |
-7 |
1 |
-6 |
3 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
- Row 1 replaced by (1/2)Row 1
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
-1/2 |
0 |
-2 |
1/2 |
| 0 |
0 |
0 |
-3 |
-6 |
-3 |
-6 |
3 |
0 |
3 |
-6 |
-3 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
3 |
| 0 |
0 |
6 |
-11 |
-7 |
1 |
-4 |
-1 |
0 |
8 |
-4 |
1 |
| 0 |
0 |
4 |
-7 |
-4 |
-1 |
4 |
-3 |
-7 |
1 |
-6 |
3 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
-
Row 4 replaced by Row 4 - 6 Row 1
Row 5 replaced by Row 5 - 4 Row 1
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
-1/2 |
0 |
-2 |
1/2 |
| 0 |
0 |
0 |
-3 |
-6 |
-3 |
-6 |
3 |
0 |
3 |
-6 |
-3 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
3 |
| 0 |
0 |
0 |
-8 |
-16 |
-8 |
-16 |
8 |
3 |
8 |
8 |
-2 |
| 0 |
0 |
0 |
-5 |
-10 |
-7 |
-4 |
3 |
-5 |
1 |
2 |
1 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
-
Row 2 replaced by (-1/3)Row 2, to make the pivot =1.
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
-1/2 |
0 |
-2 |
1/2 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
1 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
3 |
| 0 |
0 |
0 |
-8 |
-16 |
-8 |
-16 |
8 |
3 |
8 |
8 |
-2 |
| 0 |
0 |
0 |
-5 |
-10 |
-7 |
-4 |
3 |
-5 |
1 |
2 |
1 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
-
Row 1 replaced by Row 1 + (1/2)Row 2
Row 3 replaced by Row 3 - Row 2
Row 4 replaced by Row 4 + 8Row 2
Row 5 replaced by Row 5 + 5Row 2
| 0 |
0 |
1 |
0 |
5/2 |
2 |
3 |
-2 |
-1/2 |
-1/2 |
-1 |
1 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
0 |
24 |
6 |
| 0 |
0 |
0 |
0 |
0 |
-2 |
6 |
-2 |
-5 |
-4 |
12 |
6 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
| 0 |
0 |
1 |
0 |
5/2 |
2 |
3 |
-2 |
-1/2 |
-1/2 |
-1 |
1 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
1 |
| 0 |
0 |
0 |
0 |
0 |
-2 |
6 |
-2 |
-5 |
-4 |
12 |
6 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
0 |
24 |
6 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
-
Row 3 replaced by (-1/2)Row 3.
| 0 |
0 |
1 |
0 |
5/2 |
2 |
3 |
-2 |
-1/2 |
-1/2 |
-1 |
1 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
1 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
5/2 |
2 |
-6 |
-3 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
0 |
24 |
6 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
4 |
-12 |
4 |
10 |
8 |
-24 |
-12 |
-
Row 1 replaced by Row 1 -2Row 3
Row 2 replaced by Row 2 -Row 3
Row 6 replaced by Row 6 -4Row 2
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
-11/2 |
-9/2 |
11 |
7 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
-5/2 |
-3 |
8 |
4 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
5/2 |
2 |
-6 |
-3 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
0 |
24 |
6 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-
Row 4 replaced by (1/3)Row 4.
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
-11/2 |
-9/2 |
11 |
7 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
-5/2 |
-3 |
8 |
4 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
5/2 |
2 |
-6 |
-3 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
8 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-
Row 1 replaced by Row 1 +(11/2)Row 4
Row 2 replaced by Row 2 +(5/2)Row 4
Row 3 replaced by Row 3 -(5/2)Row 4
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
0 |
-9/2 |
55 |
18 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
0 |
-3 |
28 |
9 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
0 |
2 |
-26 |
-8 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
8 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-
Row 5 replaced by (1/2)Row 5.
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
0 |
-9/2 |
55 |
18 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
0 |
-3 |
28 |
9 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
0 |
2 |
-26 |
-8 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
8 |
2 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-
Row 1 replaced by Row 1 -18Row 5
Row 2 replaced by Row 2 -9Row 5
Row 3 replaced by Row 3 +8Row 5
Row 4 replaced by Row 4 -2Row 5
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
0 |
-9/2 |
55 |
0 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
0 |
-3 |
28 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
0 |
2 |
-26 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
8 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
- The resulting matrix is in the reduced echelon form now.
The five pink entries are the pivots.
REDUCED ECHELON FORM
| 0 |
0 |
1 |
0 |
5/2 |
0 |
9 |
-4 |
0 |
-9/2 |
55 |
0 |
| 0 |
0 |
0 |
1 |
2 |
0 |
5 |
-2 |
0 |
-3 |
28 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
0 |
2 |
-26 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
8 |
0 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Remark:
The reduced echelon form is unique.
The reduced echelon form is independent of the row operations you choose.
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→ |
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→ |
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→ |
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(Reduced Echelon Form) |
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Swap R1 & R3 |
→ |
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→ |
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→ |
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→ |
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(Reduced Echelon Form) |