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)
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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.
Formula
Row operations: Rᵢ ↔ Rⱼ, Rᵢ ÷ k, Rᵢ − k·Rⱼ ⇒ [I | x]
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.
Frequently asked questions
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.