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Resources on Linear Algebra

Reduced Row Echelon Form (RREF)

By Kardi Teknomo, PhD.

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There is a standard form of a row equivalent matrix that if we do a sequence of row elementary operations to reach this standard form, we may gain the solution of the linear system. The standard form is called Reduced Row Echelon Form of a matrix, or matrix RREF in short.

An m by n matrix is called to be in reduced row echelon form when it satisfies the following conditions:
1.       All zero rows, if any, are at the bottom of the matrix.
2.       Reading from left to right, the first non zero entry in each row that does not consist entirely of zeros is a 1, called the leading entry of its row.
3.       If two successive rows do not consist entirely of zeros, the second row starts with more zeros than the first (the leading entry of second row is to the right of the leading entry of first row).
4.       All other elements of the column in which the leading entry 1 occurs are zeros.

When only the first three conditions are satisfied, the matrix is called in Row Echelon Form (REF).

Using Reduced Row Echelon Form of a matrix we can calculate matrix inverse, rank of matrix, and solve simultaneous linear equations.

You will find that the educational program below is awesome. The interactive program gives many examples to compute the Reduced Row Echelon Form of a matrix input using the three row elementary operations. The computation will show you step by step through both REF and RREF.

How to use?

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

Report in rational format

Yes, this program is a free educational program!! Please don't forget to tell your friends and teacher about this awesome program!

See also: elementary row operations, matrix inverse using Gauss Jordan, rank of matrix through RREF, and solving simultaneous linear equations.

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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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