Row Reduction (4)
From an Echelon Form 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 its reduced echelon form by row operations.
ECHELON FORM
| 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 |
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 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
- Here we explain how to transform this echelon form
to "reduced echelon form".
In other words, we
- make all pivots equal to 1 (divide each pivotal row by the corresponding pivot); and
- eliminate all the nonzero entries in the pivotal columns
above the pivots.
This is essentially "backward substitution"
done in the matrix world.
- Change pivots to 1:
Row 1 replaced by (1/2)Row 1
Row 2 replaced by (-1/3)Row 2
Row 3 replaced by (-1/2)Row 3
Row 4 replaced by (1/3)Row 4
Row 5 replaced by (1/2)Row 5
| 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 |
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 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
- Next, work on the last (rightmost) pivotal column.
In this example, the working column now is Column 12.
| 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 |
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 |
1 |
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| ↓ |
|
Eliminate all nonzero entries above the pivot in Column 12: |
|
Row 1 replaced by Row 1 - (1/2)Row 5 |
|
Row 2 replaced by Row 2 - Row 5 |
|
Row 3 replaced by Row 3 + 3Row 5 |
|
Row 4 replaced by Row 4 - 2Row 5 |
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
-1/2 |
0 |
-2 |
0 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
5/2 |
2 |
-6 |
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 next working column is Column 9.
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
-1/2 |
0 |
-2 |
0 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
0 |
| 0 |
0 |
0 |
0 |
0 |
1 |
-3 |
1 |
5/2 |
2 |
-6 |
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 |
| ↓ |
|
Row 1 replaced by Row 1 + (1/2)Row 4 |
|
Row 3 replaced by Row 3 - (5/2)Row 4 |
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
0 |
0 |
2 |
0 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
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 |
| 0 |
0 |
1 |
-1/2 |
3/2 |
3/2 |
2 |
-3/2 |
0 |
0 |
2 |
0 |
| 0 |
0 |
0 |
1 |
2 |
1 |
2 |
-1 |
0 |
-1 |
2 |
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 |
| ↓ |
|
Row 1 replaced by Row 1 - (3/2)Row 3 |
|
Row 2 replaced by Row 2 - Row 3 |
| 0 |
0 |
1 |
-1/2 |
3/2 |
0 |
13/2 |
-3 |
0 |
-3 |
41 |
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 |
- Finally we work on Column 4.
| 0 |
0 |
1 |
-1/2 |
3/2 |
0 |
13/2 |
-3 |
0 |
-3 |
41 |
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 |
| ↓ |
|
Row 1 replaced by Row 1 + (1/2)Row 2 |
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 |
The five pivots are shown pink.
Remark:
In Note (3)
From a General Matrix to an Echelon Form,
we have seen that even if we start from the same matrix, different choices of row operations may give different echelon forms.
The reduced echelon form, however, 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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(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) |