# (Reduced) Row Echelon Form Calculator

URL: https://speedycalculator.com/calculators/math/row-echelon-form/
Last updated: 2026-10-30

> The row echelon form calculator reduces the augmented matrix of a 2- or 3-equation linear system to row echelon form or reduced row echelon form and gives the solution.

## Inputs

- 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?

## Results

- Transformed matrix
- Solution
- Row operations

## Formula

```
Elementary row operations: swap, scale, add a multiple of another row
```

## Explanation

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?**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### 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.

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Echelon form](https://mathworld.wolfram.com/EchelonForm.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)