Gauss-Jordan Calculator
Solve a system of linear equations using Gauss-Jordan elimination to reduced row-echelon form.
About Gauss-Jordan Calculator
Gauss-Jordan elimination transforms the augmented matrix [A|b] into reduced row-echelon form (RREF) using partial pivoting. This directly yields the solution without back-substitution. The method handles any non-singular 3×3 system. Enter coefficients for the system a₁₁x₁ + a₁₂x₂ + a₁₃x₃ = b₁, etc.
Formula
Gauss-Jordan
[A|b] → [I|x] via row operations with partial pivoting
- A
- Coefficient matrix (3×3)
- b
- Right-hand side vector
- I
- Identity matrix
- x
- Solution vector
Worked example
With the default values below, Gauss-Jordan Calculator returns x₁ of 2.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a₁₁
- 2
- a₁₂
- 1
- a₁₃
- -1
- b₁
- 8
- a₂₁
- -3
- a₂₂
- -1
- a₂₃
- 2
- b₂
- -11
- a₃₁
- -2
- a₃₂
- 1
- a₃₃
- 2
- b₃
- -3
Results
- x₁
- 2.00000000
- x₂
- 3.00000000
- x₃
- -1.00000000
Step by step
- 1Values used
- 2Gauss-Jordan
- 3x₁= 2.00000000
- 4x₂= 3.00000000
- 5x₃= -1.00000000
Assumptions and limits
- The result is derived only from the 12 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What if the system has no solution?
The calculator reports an error when the matrix is singular (determinant ≈ 0), indicating either no solution or infinitely many solutions.
Why use partial pivoting?
Partial pivoting reduces numerical errors by choosing the largest available pivot element, preventing division by near-zero values.