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Vector Triple Dot Product
Scalar Triple Product
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Matrix Transpose
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Matrix Multiplication
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Hadamard Product
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Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
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Resources on Linear Algebra

Finding inverse using RREF (Gauss-Jordan)

By Kardi Teknomo, PhD.

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We can use the three elementary row operations to find the row equivalent of Reduced Row Echelon Form (RREF) of the matrix input. In this page you will learn that using the same technique, you can obtain matrix inverse. This technique is called Gauss-Jordan method.

Gauss Jordan method is based on the following theorem:
A square matrix is invertible if and only if it is row equivalent to the identity matrix.

The method to find inverse using Gauss Jordan method is as follow:
1.       We concatenate the input matrix with identity matrix.
2.       Perform the row elementary row operations to reach RREF.
3.       If the left part of the matrix RREF is equal to an identity matrix, then the left part is the inverse matrix
4.       If the left part of the matrix RREF is not equal to an identity matrix, then we conclude the input matrix is singular. It has no inverse.

To give you example on how to compute step by step Gauss Jordan methhod, I make a matrix calculator below.

The interactive calculator below can give you as many examples as you wish to compute matrix inverse using Gauss Jordan method systematically.

Yes, this program is a free educational program!! If you like it, please recommend this fascinating program to your friends.

The program shows you step by step of the three row elementary operations to reach the Reduced Row Echelon Form of the matrix input.

How to use?

Simply click Random Example button to create new random input matrix, then click “Matrix Inverse using Gauss Jordan” button to get the whole sequence of elementary row operations from the input matrix up to the RREF and matrix inverse. The results can be in either rational or decimal format.

Report in rational format

See also: Matrix RREF, Elementary row operations

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\


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