# Gauss-Jordan Elimination Calculator

## Perform Gauss-Jordan elimination calculator step by step

The calculator will perform the Gaussian elimination on the given augmented matrix, with steps shown. Complete reduction is available optionally.

Related calculators: Reduced Row Echelon Form (RREF) Calculator, Matrix Inverse Calculator

Introducing the Gauss-Jordan Elimination Calculator—an adept and precise solution for rapidly solving systems of linear equations and converting them into their simplified Reduced Row Echelon Form (RREF). By implementing the renowned Gauss-Jordan elimination technique, a cornerstone of linear algebra, our calculator simplifies the process. It turns your system of equations into an augmented matrix and then applies a systematic series of row operations to get you the solution you need.

## How to Use the Gauss-Jordan Elimination Calculator?

### Input

On the calculator interface, you'll find several fields corresponding to the coefficients of your linear equations. Enter the numerical values of the coefficients in these fields to form your augmented matrix. Make sure you align your coefficients properly with the corresponding variables across the equations.

### Calculation

Click the "Calculate" button. The calculator will use the Gauss-Jordan method to change the matrix.

### Result

The results will be displayed automatically. The calculator will provide the resulting matrix.

## What Is Gauss-Jordan Elimination?

Gauss-Jordan elimination is an extended variant of the Gaussian elimination process. Whereas the Gaussian elimination aims to simplify a system of linear equations into a triangular matrix form to facilitate problem-solving, the Gauss-Jordan method takes it a notch higher by refining the system into a diagonal matrix, with each row standing for a unique variable. The crux of Gauss-Jordan elimination is the conversion of the matrix into what's known as its reduced row echelon form.

## Gauss-Jordan Elimination Method Explained

Let's take a quick look at the Gauss-Jordan elimination method that our calculator implements:

- Transform the system of linear equations into an augmented matrix format.
- Carry out fundamental row operations to turn the matrix into its Reduced Row Echelon Form (RREF)
- Extract the solutions straight from the resulting matrix.

For example, consider the following system of linear equations:

$$\begin{cases}2x+3y-z=9\\-x+2y+3z=8\\3x-y+2z=3\end{cases}$$Our Gauss Jordan elimination method calculator will transform this system into an augmented matrix:

$$\left[\begin{array}{ccc|c}2&3&-1&9\\-1&2&3&8\\3&-1&2&3\end{array}\right]$$By applying the Gauss-Jordan elimination algorithm, the calculator will convert this augmented matrix into its RREF, from which the solution can be read directly.

## Why Choose Our Gauss-Jordan Elimination Calculator?

### Fast and Accurate

Get quick and precise solutions for systems of linear equations. The calculator handles complex operations swiftly, providing you with accurate results in no time.

### Ease of Use

With its intuitive design, the calculator is straightforward to use. Whether you're new to the Gauss-Jordan method or an expert, you'll have no trouble getting the answers you need.

### Learning Tool

It's not just a calculator, it's also an educational resource. By providing a step-by-step breakdown of the Gauss-Jordan method, it offers a clear understanding of the process involved in solving linear equations.

### Reliable

The Gauss-Jordan calculator is based on well-established mathematical formulas, making it a reliable tool for all your linear equation solutions.

### FAQ

#### What if I face issues or have questions while using the calculator?

If you encounter any issues or have additional queries, feel free to contact us. We're always ready to assist you.

#### Can I use the calculator to check my manual calculations?

Absolutely! Our Gauss-Jordan Elimination Calculator is an excellent tool to validate your manual calculations, providing an efficient way to cross-check your answers.

#### How accurate is the Gauss-Jordan Elimination Calculator?

Our calculator is built on the established mathematical principles of the Gauss-Jordan elimination method, ensuring correct and precise results.

#### Can I use this calculator for any system of linear equations?

The calculator is designed to handle any system. But if the number of equations doesn't equal the number of variables, there will be either an infinite number of solutions or no solution at all.