Row echelon form is the staircase shape that Gaussian elimination produces; reduced row echelon form goes further so that each pivot is 1 and the only non-zero entry in its column. From it you can read the solution, the rank and the free variables.
How to use the (Reduced) Row Echelon Form Calculator
- Number of equations
- a₁ (x)
- b₁ (y)
- c₁ (z)
- d₁ (right side)
- a₂ (x)
- b₂ (y)
- c₂ (z)
- d₂ (right side)
- a₃ (x)
- b₃ (y)
- c₃ (z)
- d₃ (right side)
- Show reduced form?
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How it works
Choose whether you want the reduced form. Row echelon form is reached by eliminating below each pivot; the reduced form also eliminates above and scales pivots to 1.
Formula
Elementary row operations: swap, scale, add a multiple of another row
Worked example
Inputs — Number of equations: Three equations (x, y, z); a₁ (x): 2; b₁ (y): 1; c₁ (z): -1; d₁ (right side): 8; a₂ (x): -3; b₂ (y): -1; c₂ (z): 2; d₂ (right side): -11; a₃ (x): -2; b₃ (y): 1; c₃ (z): 2; d₃ (right side): -3; Show reduced form?: Yes – reduced row echelon form.
Results — Transformed matrix = ⎡ 1 0 0 2 ⎤
⎢ 0 1 0 3 ⎥
⎣ 0 0 1 −1 ⎦; Solution = x = 2, y = 3, z = −1; Row operations = R1 ↔ R2
R1 ÷ −3
R2 − (2)·R1
R3 − (−2)·R1
R2 ↔ R3
R2 ÷ 5/3
R1 − (1/3)·R2
R3 − (1/3)·R2
R3 ÷ 1/5
R1 − (−4/5)·R3
R2 − (2/5)·R3.
Key terms
- Pivot
- The leading non-zero entry of a row.
Frequently asked questions
What is row echelon form?
A staircase pattern with zeros below each leading entry.
What makes it "reduced"?
Each pivot is 1 and is the only non-zero entry in its column.
Is row echelon form unique?
No, but reduced row echelon form is.
How do I read the solution?
From the reduced form, each pivot row gives one variable.