Gauss Elimination Calculator
Gaussian elimination reduces the system to triangular form and back-substitutes: this classic 3 × 3 system has x1 = 2, x2 = 3, x3 = -1.
For 2x + y − z = 8 enter “2 1 -1 8”.
x1 = 2, x2 = 3, x3 = -1
- x1 (decimal)
- 2
- x2 (decimal)
- 3
- x3 (decimal)
- -1
| x1 | x2 | x3 | b | |
|---|---|---|---|---|
| row 1 | -3 | -1 | 2 | -11 |
| row 2 | 0 | 5/3 | 2/3 | 13/3 |
| row 3 | 0 | 0 | 1/5 | -1/5 |
Step-by-step solution
- Swap R1 and R2 (partial pivoting)
- R2 → R2 − (-2/3)·R1
- R3 → R3 − (2/3)·R1
- Swap R2 and R3 (partial pivoting)
- R3 → R3 − (1/5)·R2
- Back substitution from the last row upwards
Gauss Elimination Formula
Rᵢ → Rᵢ − (aᵢₖ/aₖₖ)·Rₖ
forward elimination to upper triangular form, then back substitution
Worked Example
The system 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3:
- Swap R1 and R2 (partial pivoting)
- R2 → R2 − (-2/3)·R1
- R3 → R3 − (2/3)·R1
- Swap R2 and R3 (partial pivoting)
- R3 → R3 − (1/5)·R2
- Back substitution from the last row upwards
x2 (decimal): 3 · x3 (decimal): -1
| x1 | x2 | x3 | b | |
|---|---|---|---|---|
| row 1 | -3 | -1 | 2 | -11 |
| row 2 | 0 | 5/3 | 2/3 | 13/3 |
| row 3 | 0 | 0 | 1/5 | -1/5 |
GATE tip
If elimination produces a row 0 = c with c ≠ 0, the system has no solution; a row of all zeros means infinitely many. The matrix rank calculator gives the verdict directly.
More Linear Algebra Calculators
-
Matrix Rank Calculator
rank = number of non-zero rows in echelon form
unique if rank A = rank [A|b] = n; none if rank A < rank [A|b]
Example: The matrix [[1, 2, 3], [2, 4, 6], [1, 0, 1]]: Rank = 2.
-
LU Decomposition Calculator
A = LU, uᵢⱼ = aᵢⱼ − Σ lᵢₖuₖⱼ, lⱼᵢ = (aⱼᵢ − Σ lⱼₖuₖᵢ)/uᵢᵢ
Doolittle: L has 1s on the diagonal
Example: The matrix [[2, −1, −2], [−4, 6, 3], [−4, −2, 8]]: det A = product of U’s diagonal = 24.
Frequently asked questions
What is partial pivoting?
Before eliminating a column, swap in the row with the largest absolute value in that column. It avoids dividing by small numbers and reduces rounding error.
How many operations does Gauss elimination need?
About n³/3 multiplications for n equations, which is why it is preferred over Cramer’s rule for more than three unknowns.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Gauss Elimination Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/gauss-elimination-calculator/
IEEE
Scientific Converters, "Gauss Elimination Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/gauss-elimination-calculator/