Gaussian Elimination Calculator

Enter Matrix Coefficients

Use the controls below to solve a square system with two to eight unknowns.

Example Data Table

This sample system matches the built-in example button.

Equation	x Coefficient	y Coefficient	z Coefficient	Constant
1	2	1	-1	8
2	-3	-1	2	-11
3	-2	1	2	-3

Expected solution: x = 2, y = 3, z = -1.

Formula Used

Forward elimination multiplier: m_ik = a_ik / a_kk

Row operation: R_i ← R_i − m_ikR_k

Back substitution: x_i = (b_i − Σ a_ijx_j) / a_ii

Determinant: det(A) = (−1)^s × Π pivots, where s is row swaps.

Gaussian elimination transforms the augmented matrix into upper triangular form. Then back substitution solves each variable from the last row upward.

The calculator also estimates rank by counting nonzero rows after elimination. If rank(A) differs from rank([A|b]), the system is inconsistent.

Residual checking uses r = Ax − b. A small residual confirms that the computed solution numerically matches the original system.

How to Use This Calculator

Choose the system size from 2 × 2 up to 8 × 8.

Enter all matrix coefficients and the constant term for each equation.

Set decimal precision and a numerical tolerance that suits your data.

Enable partial pivoting for improved stability when pivots are very small.

Click Solve System to view status, solution, determinant, ranks, matrix steps, and the graph.

Use the export buttons after calculation to download a CSV or PDF summary.

FAQs

1. What does Gaussian elimination do?

It converts a linear system into an upper triangular matrix using row operations. That structure makes back substitution easy, so each variable can be solved from the bottom row upward.

2. Why is pivoting useful?

Pivoting selects a stronger pivot value before elimination. This reduces division by tiny numbers, improves stability, and often produces more reliable answers for ill-conditioned systems.

3. What does determinant zero mean here?

A zero determinant means the coefficient matrix is singular. The system may have either no solution or infinitely many solutions, depending on the augmented column.

4. How are ranks used?

Rank compares the number of independent rows in the coefficient matrix and augmented matrix. Equal ranks indicate consistency. Unequal ranks indicate contradiction and no valid solution.

5. What is a residual?

The residual is Ax − b after substituting the computed solution into the original equations. Smaller residuals suggest that rounding and numerical error stayed under control.

6. Can this calculator detect infinite solutions?

Yes. When rank(A) equals rank([A|b]) but both are smaller than the number of variables, the system is dependent and has infinitely many solutions.

7. When should I change tolerance?

Adjust tolerance when your matrix contains very small values or rounding noise. A slightly larger tolerance can help classify nearly zero entries more sensibly.

8. What does the graph show?

The chart visualizes pivot magnitudes and, when available, solution values. It helps you quickly inspect scaling, stability, and the numeric pattern across elimination stages.