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Gaussian Elimination Calculator

Set the matrix of a linear equation and write down entries of it to determine the solution by applying the Gaussian elimination method using this calculator.

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Gaussian Elimination Calculator

This Gaussian elimination calculator allows you to solve systems of linear equations efficiently and accurately. Whether you are working on homework or complex matrix problems, this tool simplifies the entire process.

What is the Gaussian Elimination Algorithm?

Gaussian elimination is a systematic procedure used to solve systems of linear equations. It works by transforming the system’s augmented matrix into row echelon form (or reduced row echelon form) using elementary row operations.

How to Use the Gaussian Elimination Method

Let’s understand the process step by step with a clear example.

Example #1:

Solve the following system of equations:

$$ 2x_1 + 4x_2 = 18 \\ 4x_1 + 6x_2 = 30 $$

Solution:

First, write the system in augmented matrix form:

$$ \begin{bmatrix} 2 & 4 & 18 \\ 4 & 6 & 30 \end{bmatrix} $$

Gaussian Elimination Steps:

Step 1: Divide the first row by 2 to make the leading coefficient 1:

$$ \begin{bmatrix} 1 & 2 & 9 \\ 4 & 6 & 30 \end{bmatrix} $$

Step 2: Eliminate the first variable from the second row by subtracting 4 times the first row from the second row:

$$ \begin{bmatrix} 1 & 2 & 9 \\ 0 & -2 & -6 \end{bmatrix} $$

Step 3: Divide the second row by -2 to get a leading 1:

$$ \begin{bmatrix} 1 & 2 & 9 \\ 0 & 1 & 3 \end{bmatrix} $$

Step 4: Eliminate the second variable from the first row by subtracting 2 times the second row from the first row:

$$ \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 3 \end{bmatrix} $$

Now the left side of the matrix is the identity matrix. Therefore, the solution is:

$$ x_1 = 3, \quad x_2 = 3 $$

You can check your answer instantly using our free online Gaussian elimination calculator.

How the Gaussian Elimination Calculator Works

Input:

Select the matrix size (number of variables) from the drop-down menu.
Click Set Matrix to generate the input fields.
Enter the coefficients and constant values in the provided boxes.
Press Calculate to compute the solution.

Output:

Displays the final values of all variables.
Provides a detailed step-by-step breakdown of the elimination process.

References:

Wikipedia: Gaussian Elimination – Explanation of row operations, echelon forms, determinants, and matrix inversion.

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