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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Finding inverse using RREF (Gauss-Jordan)

By Kardi Teknomo, PhD.
LinearAlgebra

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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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