Calculator
Enter A and b for the system Ax = b. Choose size, precision, and pivoting options. Then press Solve.
Example Data Table
Sample 3×3 system. Use “Fill example” to load it into the form.
| A1 | A2 | A3 | b |
|---|---|---|---|
| 2 | 1 | -1 | 8 |
| -3 | -1 | 2 | -11 |
| -2 | 1 | 2 | -3 |
Formula Used
- Augmented matrix: Write the system as [A|b].
- Row operations: Swap rows, scale rows, and add multiples of one row to another.
- Elimination: Create zeros below each pivot to form an upper-triangular matrix.
- Back substitution: Solve from the last row upward to compute each variable.
- Rank test: Compare rank(A) and rank([A|b]) to detect none/unique/infinite solutions.
- Determinant: For full rank, det(A) equals product of pivots times sign from swaps.
How to Use This Calculator
- Select the system size n and your preferred precision.
- Enter the coefficients of matrix A and constants vector b.
- Keep partial pivoting enabled for better numerical stability.
- Enable steps if you want to study each row operation.
- Press Solve to see results above the form.
- Use the download buttons to export CSV or PDF reports.
FAQs
1) What does Gaussian elimination solve?
It solves linear equation systems by turning the matrix into an upper-triangular form, then computing variables using back substitution.
2) Why is partial pivoting recommended?
Pivoting swaps in a larger pivot value, reducing division by tiny numbers. This improves numerical stability and often produces more reliable answers.
3) What if the calculator says “No solution”?
That means the elimination produced a row like 0 = nonzero. The equations contradict each other, so no vector x can satisfy every equation.
4) What does “Infinitely many solutions” mean?
It means rank(A) is less than the number of variables, so at least one variable is free. The calculator outputs one valid solution by setting free variables to zero.
5) How should I enter fractions?
Enter them as decimals, like 0.25 for 1/4. The solver works with floating-point arithmetic and rounds displayed results to your chosen precision.
6) What is the residual check?
It computes r = Ax − b using your original inputs. A small maximum |r| indicates the solution fits the system well within rounding error.
7) Is the determinant always shown?
If the matrix is full rank, the determinant is computed from the triangular pivots and row swaps. If rank(A) is smaller, the determinant is reported as zero.