Gauss Elimination Calculator Step by Step

Calculator Input

Example Data Table

Use this sample to test the calculator before entering your own system.

Formula Used

Gaussian elimination converts a system into upper triangular form. The augmented matrix is written as [A|b].

Elimination factor:

mᵢₖ = aᵢₖ / aₖₖ

Row operation:

Rᵢ ← Rᵢ - mᵢₖRₖ

Back substitution:

xᵢ = (bᵢ - Σ aᵢⱼxⱼ) / aᵢᵢ

Determinant after elimination:

det(A) = (-1)ˢ × product of pivots

Here, s is the number of row swaps. This applies because the calculator does not scale pivot rows during elimination.

How to Use This Calculator

1. Select the number of equations and variables.
2. Enter one augmented matrix row per line.
3. Separate values with commas, spaces, or semicolons.
4. Use fractions like 3/4 when needed.
5. Choose decimal precision and pivoting mode.
6. Press Calculate to view the answer and steps.
7. Use CSV or PDF buttons to save the report.

Gauss Elimination Guide

What Is Gaussian Elimination?

Gaussian elimination is a core method for solving linear systems. It changes equations into a simpler upper triangular form. The solution is then found by back substitution. This calculator shows each row operation, so the process stays transparent.

How the Matrix Is Prepared

The method starts with an augmented matrix. Each equation becomes one row. Coefficients are placed on the left. Constants are placed on the right. A pivot is selected in the first useful column. Rows below that pivot are cleared by adding a multiple of the pivot row.

Why Pivoting Matters

Partial pivoting can improve stability. It swaps the current row with a lower row that has a larger pivot value. This reduces division by very small numbers. It also helps detect singular systems. A singular system cannot produce one unique solution.

Back Substitution

After forward elimination, the matrix has zeros below the main diagonal. Back substitution begins with the last equation. Each known value is moved into the equations above it. The calculator reports every variable with the selected precision. It also lists the final augmented matrix.

Learning Value

Step by step output is useful for learning. You can compare manual work with the generated operations. The row labels make it easier to find mistakes. Each operation follows the same structure. A row is swapped, scaled, or replaced with a row combination.

Input Rules

This tool supports square systems from two to six variables. Enter one augmented row per line. Separate numbers with spaces or commas. Use decimals or fractions like 3/4. Keep every row the same length. The last value in each row is the constant term.

Saving Results

Use the CSV export to save a spreadsheet friendly summary. Use the PDF export to keep a printable report. The example table gives ready test data. Try it first, then replace it with your own system.

Practical Use

Gaussian elimination is widely used in engineering, economics, physics, and data work. It handles many practical models where several unknowns interact. The calculator does not replace understanding. It supports it by showing the logic behind each answer.

Accuracy Notes

When results seem unexpected, review the input rows first. Small entry errors change the entire solution. Also check whether pivoting is enabled. For near singular matrices, compare rounded answers with higher precision. This helps judge sensitivity before using it safely.

FAQs