# Gauss-Jordan Elimination Calculator

URL: https://speedycalculator.com/calculators/math/gauss-jordan-elimination/
Last updated: 2026-10-30

> The Gauss–Jordan elimination calculator solves a system of two or three linear equations by reducing the augmented matrix to reduced row echelon form, listing every row operation.

## 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)

## Results

- Solution
- x
- y
- z
- Augmented matrix
- Reduced row echelon form
- Row operations

## Formula

```
Row operations: Rᵢ ↔ Rⱼ, Rᵢ ÷ k, Rᵢ − k·Rⱼ  ⇒  [I | x]
```

## Explanation

Gauss–Jordan elimination is the systematic version of solving equations by elimination. Once the augmented matrix is in reduced row echelon form, the solution can be read directly – or the method reveals that there is no solution or infinitely many.

### How to use the Gauss-Jordan Elimination 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)**

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

### How it works

For each column a pivot is chosen (largest absolute value, for stability), its row is scaled to make the pivot 1 and multiples are subtracted from every other row. Integer systems are worked with exact fractions.

## 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.

Results — **Solution = x = 2, y = 3, z = −1**; x = 2; y = 3; z = −1; Augmented matrix = ⎡ 2 1 −1 8 ⎤
⎢ −3 −1 2 −11 ⎥
⎣ −2 1 2 −3 ⎦; Reduced row echelon form = ⎡ 1 0 0 2 ⎤
⎢ 0 1 0 3 ⎥
⎣ 0 0 1 −1 ⎦.

## Key terms

- **Pivot**: The leading non-zero entry used to eliminate others.

## FAQ

**What is the difference between Gaussian and Gauss–Jordan elimination?**
Gaussian stops at row echelon form and back-substitutes; Gauss–Jordan continues to reduced form.

**What does a row of zeros mean?**
A dependent equation – there may be infinitely many solutions.

**What does [0 0 0**
1] mean? | The system is inconsistent – no solution.

**Why choose the largest pivot?**
Partial pivoting reduces rounding errors.


## Sources

- [Wolfram MathWorld – Gauss-Jordan elimination](https://mathworld.wolfram.com/Gauss-JordanElimination.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/)