Gaussian elimination calculator with steps
Enter the coefficients and solve. You see every row operation and the matrix after it, exact with fractions. It also tells you when there is no solution or infinitely many. Or work out a determinant.
Size
Result
x₁ = 2
x₂ = 3
x₃ = −1
Steps
Starting matrix
| 2 | 1 | −1 | 8 |
| −3 | −1 | 2 | −11 |
| −2 | 1 | 2 | −3 |
12·II + 3·I
| 2 | 1 | −1 | 8 |
| 0 | 1 | 1 | 2 |
| −2 | 1 | 2 | −3 |
2III + I
| 2 | 1 | −1 | 8 |
| 0 | 1 | 1 | 2 |
| 0 | 2 | 1 | 5 |
3III − 2·II
| 2 | 1 | −1 | 8 |
| 0 | 1 | 1 | 2 |
| 0 | 0 | −1 | 1 |
4−III
| 2 | 1 | −1 | 8 |
| 0 | 1 | 1 | 2 |
| 0 | 0 | 1 | −1 |
5I + III
| 2 | 1 | 0 | 7 |
| 0 | 1 | 1 | 2 |
| 0 | 0 | 1 | −1 |
6II − III
| 2 | 1 | 0 | 7 |
| 0 | 1 | 0 | 3 |
| 0 | 0 | 1 | −1 |
7I − II
| 2 | 0 | 0 | 4 |
| 0 | 1 | 0 | 3 |
| 0 | 0 | 1 | −1 |
81/2·I
| 1 | 0 | 0 | 2 |
| 0 | 1 | 0 | 3 |
| 0 | 0 | 1 | −1 |
How Gaussian elimination works
- Write the system as an augmented matrix.
- Use row operations to make zeros below every pivot (row echelon form).
- Back-substitute, or keep going until the left side is the identity (Gauss-Jordan).
Determinant: every row swap flips the sign; the determinant is then the product of the diagonal.
Frequently asked questions
My steps look different. Are they wrong?+
Not necessarily. There are many correct paths; the solution at the end must be the same.