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Row Reduction (RREF)
Row Reduction (RREF)
Reduce a matrix to row echelon form, step by step
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Rows
2
3
4
Columns
2
3
4
5
Augmented system (last column = constants)
Matrix
rref(A) · Rank = 3 · Unique solution
Reduced row echelon form
1
0
0
2
0
1
0
3
0
0
1
−1
Rank
3
Nullity
0
Pivot columns
1, 2, 3
Free columns
none
Linear system
Unique solution
Steps
1.
Scale row
R1 → 1/2·R1
1
1/2
−1/2
4
−3
−1
2
−11
−2
1
2
−3
2.
Eliminate
R2 → R2 + 3·R1
1
1/2
−1/2
4
0
1/2
1/2
1
−2
1
2
−3
3.
Eliminate
R3 → R3 + 2·R1
1
1/2
−1/2
4
0
1/2
1/2
1
0
2
1
5
4.
Scale row
R2 → 2·R2
1
1/2
−1/2
4
0
1
1
2
0
2
1
5
5.
Eliminate
R1 → R1 − 1/2·R2
1
0
−1
3
0
1
1
2
0
2
1
5
6.
Eliminate
R3 → R3 − 2·R2
1
0
−1
3
0
1
1
2
0
0
−1
1
7.
Scale row
R3 → -1·R3
1
0
−1
3
0
1
1
2
0
0
1
−1
8.
Eliminate
R1 → R1 + 1·R3
1
0
0
2
0
1
1
2
0
0
1
−1
9.
Eliminate
R2 → R2 − 1·R3
1
0
0
2
0
1
0
3
0
0
1
−1
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Original matrix reduced to RREF with pivot positions highlighted
The input matrix on the left is transformed into its reduced row echelon form on the right; the leading 1 of each pivot column is highlighted.
2
1
−1
8
−3
−1
2
−11
−2
1
2
−3
rref
1
0
0
2
0
1
0
3
0
0
1
−1
