Gauss Jordan Elimination Calculator With Steps

Calculator

Operation

Decimal Precision

Pivot Option

Use partial pivoting

Matrix A or Augmented Matrix

Use one row per line. Separate values with spaces or commas.

Vector b

Required only when solving Ax = b.

Sample Controls

Example Data Table

Mode	Matrix Input	Vector Input	Expected Output
Solve Ax = b	2 1 -1 -3 -1 2 -2 1 2	8 -11 -3	x1 = 2, x2 = 3, x3 = -1
RREF	1 2 1 4 2 4 0 6 3 6 3 12	Not needed	Reduced row echelon form with rank details
Inverse	4 7 2 6	Not needed	Inverse matrix shown step by step
Determinant	3 2 1 1 0 2 4 1 5	Not needed	Single determinant value

Formula Used

Gauss Jordan elimination uses elementary row operations to transform a matrix into reduced row echelon form.

Allowed operations are R_i ↔ R_j, R_i → kR_i, and R_i → R_i + kR_j.

For solving a system, the calculator builds [A|b] and reduces it until pivot columns identify the variables.

For inverse mode, it builds [A|I]. When the left side becomes I, the right side becomes A^-1.

For determinant mode, row swaps change the sign, and triangular pivots are multiplied.

How to Use This Calculator

Select the required operation.
Enter one matrix row per line.
Use commas or spaces between values.
Enter vector b only for solve mode.
Choose decimal precision and pivoting settings.
Press Calculate to view the answer above the form.
Use CSV or PDF export after a valid result appears.

Gauss Jordan Elimination Calculator Overview

This Gauss Jordan elimination calculator helps students, teachers, and engineers reduce matrices without doing every row operation by hand. It accepts coefficient matrices, augmented matrices, and identity augmentation for inverse work. The tool shows each pivot choice, row swap, scaling action, and row replacement. That makes the answer easier to audit.

Why Row Reduction Matters

Gauss Jordan elimination converts a matrix into reduced row echelon form. In that form, leading entries become one. All other entries in a pivot column become zero. These simple rules reveal many facts. You can solve a linear system. You can detect no solution. You can spot infinitely many solutions. You can also find inverse matrices when the input matrix is square and nonsingular.

What This Calculator Includes

The calculator supports solving Ax = b, reducing any entered matrix, finding determinants, and calculating inverse matrices. It also accepts decimal values and fraction values. Precision control helps you round final displays without changing the internal process. Partial pivoting improves stability when a pivot is small. The step log keeps the work transparent.

How Results Should Be Read

A unique solution appears when every variable column has a pivot and the augmented column does not create a contradiction. No solution appears when a row reduces to zero coefficients with a nonzero constant. Infinite solutions appear when at least one variable has no pivot and the system stays consistent. For inverse mode, the right side of the final augmented matrix becomes the inverse.

Good Input Practices

Keep every row the same length. Put one row on each line. Use commas, spaces, or semicolons as separators. For solve mode, enter A in the matrix box and b in the vector box. For RREF mode, you may enter an augmented matrix directly. Check units and signs before pressing calculate. A single wrong sign can change the entire solution.

Exporting and Learning

The CSV export is useful for spreadsheets. The PDF export is useful for reports. Both include the operation, input, final result, and step summary. Use the example table to test the calculator first. Then replace the sample values with your own problem. The printed steps can support homework review, classroom demonstrations, and checks before exams.

FAQs

What is Gauss Jordan elimination?

It is a row reduction method. It turns a matrix into reduced row echelon form. That final form makes solutions, rank, inverse matrices, and consistency easier to identify.

Can this calculator solve three variable systems?

Yes. Enter a 3 by 3 coefficient matrix and a three value vector. The calculator reduces the augmented matrix and displays each pivot operation.

Can I enter fractions?

Yes. Values like 1/2, -3/4, and 2.5/5 are accepted. The calculator converts them into decimal values for calculation and display.

What does partial pivoting do?

Partial pivoting chooses the largest available pivot in a column. This reduces numerical error and helps avoid division by very small values.

When does no solution appear?

No solution appears when the reduced augmented matrix contains a row with zero coefficients and a nonzero constant. That row creates a contradiction.

When are infinitely many solutions possible?

Infinite solutions appear when the system is consistent and at least one variable has no pivot. That variable becomes a free parameter.

Can this calculator find an inverse matrix?

Yes. Choose inverse mode and enter a square matrix. The calculator augments it with the identity matrix and reduces it step by step.

What is included in downloads?

The downloads include the selected operation, input matrix, summary, final result, and step notes. CSV is useful for sheets. PDF is useful for sharing.