# Gauss Jordan elimination calculator with steps | Augmented matrix calculator

## Gauss Jordan elimination calculator

The Gauss Jordan Elimination Calculator presented here enables the solution of any type of system of linear equations using the Gauss-Jordan elimination method. Despite its simplicity, the online Gauss Jordan calculator provides detailed explanations of solutions through its user-friendly interface, making it an excellent tool for studying this method of solving systems of linear equations.

To use this calculator, you just need to follow these simple steps:

1. Adjust the input matrix of the calculator according to the number of equations in the system you want to solve, using the “+” and “-” buttons as appropriate.
2. Enter the coefficients corresponding to each of the equations in the input fields of the calculator.
3. Indicate the level of precision you desire by specifying the number of decimal places to be considered.
4. Finally, click the “Calculate” button and the detailed step-by-step solution will be displayed automatically.

The Gauss-Jordan elimination calculator is also known as an augmented matrix calculator because it takes as input the augmented matrix of a system of linear equations and performs the necessary row operations to transform it into reduced row echelon form, allowing the user to easily read off the solutions to the system.

## What is Gauss Jordan elimination?

Gauss Jordan elimination is a mathematical algorithm used to solve systems of linear equations. It involves manipulating a matrix of coefficients representing the equations using row operations such as adding or subtracting multiples of one row to another or multiplying a row by a constant. The goal is to transform the matrix into a simplified form called the reduced row echelon form, which allows for the easy identification of the solution to the system of equations. The process is named after mathematicians Carl Friedrich Gauss and Wilhelm Jordan, who independently developed it in the early 19th century.

## How to do Gauss Jordan elimination

To perform Gauss Jordan elimination, we first represent the system of linear equations as an augmented matrix. We then use row operations, such as multiplying a row by a constant, swapping two rows, or adding or subtracting a multiple of one row to another, to transform the matrix into a simpler form called the reduced row echelon form.

The reduced row echelon form has several properties that make it easy to identify the solution to the system of equations. First, each non-zero row has a leading entry of 1, which is the only non-zero entry in its column. Second, the leading entry of each non-zero row is to the right of the leading entry of the row above it. Finally, any rows of zeros are at the bottom of the matrix.

Once we have the matrix in reduced row echelon form, we can read off the solution to the system of equations directly. If a row corresponds to an equation with a unique solution, then the rightmost entry of that row gives the value of the variable corresponding to the last column. If a row corresponds to an equation with infinitely many solutions, then the variables corresponding to the columns with leading entries are free, and we can express the remaining variables in terms of them.

Now we will proceed to present how to apply step by step the Gauss-Jordan elimination method through the following example:

Example: Use Gauss Jordan elimination to solve the following linear system

 5x+3y = 8 12x−9y = 41

Solution by Gauss-Jordan elimination

$\underset{}{\left(\begin{array}{ccc}\overline{)5}& 3& 8\\ 12& -9& 41\end{array}\right)}$
$×\left(0.2\right)$
$\stackrel{}{\underset{{R}_{1}∕\left(5\right)\to {R}_{1}}{~}}$$\underset{}{\left(\begin{array}{ccc}\overline{)1}& 0.6& 1.6\\ 12& -9& 41\end{array}\right)}$
$×\left(-12\right)$
$\stackrel{}{\underset{{R}_{2}-12\cdot {R}_{1}\to {R}_{2}}{~}}$$\underset{}{\left(\begin{array}{ccc}1& 0.6& 1.6\\ 0& \overline{)-16.2}& 21.8\end{array}\right)}$
$×\left(-0.062\right)$
$\stackrel{}{\underset{{R}_{2}∕\left(-16.2\right)\to {R}_{2}}{~}}$$\underset{}{\left(\begin{array}{ccc}1& 0.6& 1.6\\ 0& \overline{)1}& -1.346\end{array}\right)}$
$×\left(-0.6\right)$
$\stackrel{}{\underset{{R}_{1}-\left(0.6\right)\cdot {R}_{2}\to {R}_{1}}{~}}$$\underset{}{\left(\begin{array}{ccc}1& 0& 2.407\\ 0& 1& -1.346\end{array}\right)}$
• Find the variable $y$ from the equation $2$ of the system :
$y=-1.346$
• Find the variable $x$ from the equation $1$ of the system :
$x=2.407$
• $x=2.407$
• $y=-1.346$
General Solution: $X=\underset{}{\left(\begin{array}{c}2.407\\ -1.346\end{array}\right)}$

## Applications

Here are the most common applications of this method:

1. Solving Systems of Linear Equations: The primary application of the Gauss-Jordan elimination method is to solve systems of linear equations. By performing row operations on the augmented matrix, we can transform it into reduced row echelon form, which provides us with a way to directly read off the solutions to the system.

2. Finding the Rank of a Matrix: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. The Gauss-Jordan elimination method can be used to find the rank of a matrix by transforming it into reduced row echelon form and counting the number of nonzero rows.

3. Finding the Inverse of a Matrix: The Gauss-Jordan elimination method can also be used to find the inverse of a matrix. We start by augmenting the matrix with the identity matrix, and then perform row operations on the augmented matrix until the left half becomes the identity matrix. The right half of the augmented matrix will then be the inverse of the original matrix (if it exists).

4. Solving Homogeneous Systems of Linear Equations: A homogeneous system of linear equations is one where all the constants on the right-hand side of the equations are zero. The Gauss-Jordan elimination method can be used to determine whether a homogeneous system has nontrivial solutions (i.e., solutions other than the trivial solution where all variables are zero). If the transformed matrix has a row of zeros with a nonzero constant on the right-hand side, then the system has nontrivial solutions.

5. Solving Nonhomogeneous Systems of Linear Equations: The Gauss-Jordan elimination method can also be used to solve nonhomogeneous systems of linear equations by first solving the associated homogeneous system (i.e., the system obtained by setting all constants on the right-hand side to zero), and then adding a particular solution to the homogeneous solution to get the general solution.