What is Gaussian Jordan Elimination?

It is named after Carl Friedrich Gauss. This method works by converting a coefficient matrix into row echelon form using row swapping, row scaling, and row addition. After that, the simplified matrix is used to determine the solution of the system through back substitution or direct evaluation.

How to Perform Gaussian Elimination

In order to perform Gaussian elimination, the equation is first written as an augmented matrix. The primary focus is to create zeros below every pivot element by applying elementary row operations. This is referred to as forward elimination. Once the row echelon is formed, back substitution is applied to get the values of the variables.

Example

Let’s perform Gaussian Elimination on the following matrix:

Solution

Subtract row 1 multiplied by 7 from row 2:

R2=R2−7R1

Divide row 2 by −22:

R2=R2/-22

Subtract row 2 multiplied by 3 from row 1:

R1=R1-3R2

Answer

The resulting matrix is in row echelon form:

So, the solution of the system is:

x = 2

y = 4

The above examples show how Gaussian elimination reduces complex equations into a simplified form that can be solved easily.

Gauss-Jordan Elimination Calculator

This method becomes useful when you want to solve linear equations in a direct way. In contrast to the normal Gaussian elimination, this method reduces the matrix completely into row echelon form, in which every variable is isolated automatically. This makes the final solution easily readable directly from the matrix without the need to apply back substitution.

The following is a demonstration of this method using a linear equation:

x + y = 4

x − y = 2

Step 1: Write the Augmented Matrix

First, convert the system into an augmented matrix

Each row represents an equation, and each column represents a variable or constant.

Step 2: Convert the Matrix to Reduced Row Echelon Form

Subtract Row 1 from Row 2 to eliminate the x term in the second row:

R₂ = R₂ − R₁

Divide Row 2 by −2 to make the pivot element equal to 1:

R₂ = R₂ ÷ (−2)

Eliminate the value above the second pivot by subtracting Row 2 from Row 1:

R₁ = R₁ − R₂

The matrix is now in reduced row echelon form.

Final Answer

x = 3

y = 1

Matrix operations are widely used in advanced mathematics. If you need to work with determinants and transformations of multivariable functions, you can also try our Jacobian Calculator.

Gaussian Elimination FAQs

What is a Gaussian Elimination Calculator?

A Gaussian Elimination Calculator solves systems of linear equations step by step using matrix row operations for accurate and clear results.

What operations are used in Gaussian Elimination?

It uses row swapping, row scaling, and row addition or subtraction to simplify matrices.

What is row echelon form?

Row echelon form is a matrix structure where zeros appear below each pivot element, making equations easier to solve.

What is an augmented matrix?

An augmented matrix represents a system of linear equations, including constants in the final column.

Gauss Jordan Elimination Calculator​

Gauss Jordan Elimination Calculator