Build and reduce an augmented matrix
Enter decimals or fractions like 1/2. Results appear above this form after submission.
Example data table
This example demonstrates a solvable 3 x 3 system with a unique answer.
| Equation row | x1 | x2 | x3 | Constant |
|---|---|---|---|---|
| Row 1 | 2 | 1 | -1 | 8 |
| Row 2 | -3 | -1 | 2 | -11 |
| Row 3 | -2 | 1 | 2 | -3 |
| Expected solution | 2 | 3 | -1 | Verified |
Formula used
Gauss Jordan elimination transforms the augmented matrix [A|b] into reduced row echelon form by applying three valid row operations.
- Row swap: Ri ↔ Rj
- Row scaling: Ri ← Ri / pivot
- Row replacement: Rj ← Rj - kRi
After reduction, the solution test uses rank relationships.
- Unique solution when rank(A) = rank([A|b]) = n
- Infinite solutions when rank(A) = rank([A|b]) < n
- No solution when rank(A) < rank([A|b])
How to use this calculator
- Choose the number of variables and equations.
- Enter every coefficient and the right-side constant.
- Use integers, decimals, or fractions like 3/5.
- Pick the display precision that suits your work.
- Press Solve Matrix to reduce the system.
- Review the result panel above the form.
- Check ranks, determinant, inverse, and stored row steps.
- Export the report using the CSV or PDF buttons.
FAQs
1. What does Gauss Jordan elimination do?
It reduces an augmented matrix to reduced row echelon form. That final form reveals whether a linear system has one solution, many solutions, or no solution.
2. Can I enter fractions instead of decimals?
Yes. The calculator accepts common fraction formats such as 1/2, -3/4, and 7/5. Blank fields are treated as zero.
3. Why does the calculator show infinite solutions?
That happens when the coefficient rank matches the augmented rank, but both ranks stay below the variable count. At least one variable becomes free.
4. What causes a no-solution result?
A contradictory row appears after reduction, usually a row of zeros equaling a nonzero constant. That means the equations conflict and cannot be satisfied together.
5. Why is the determinant useful here?
For a square coefficient matrix, a nonzero determinant confirms invertibility and a unique solution. A zero determinant signals singular behavior and possible dependence.
6. Does the calculator compute the inverse matrix?
Yes, when the coefficient matrix is invertible. If the determinant is zero, the inverse section stays hidden because no valid inverse exists.
7. What does the residual check mean?
It substitutes the solved values back into each original equation. Small or zero residuals show that the computed solution satisfies the entered system.
8. What does the Plotly graph compare?
It compares row magnitudes before elimination and after reduction. This offers a visual summary of how the matrix structure changes during the process.