Gauss-Jordan Calculator
Solve a system of linear equations using Gauss-Jordan elimination to reduced row-echelon form.
Inputs
x₁
2.00000000
x₂
3.00000000
x₃
-1.00000000
Step by step
Values 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
Gauss-Jordan
[A|b] → [I|x] via row operations with partial pivoting
x₁
= 2.00000000
x₂
= 3.00000000
x₃
= -1.00000000
How it works
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
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.