Difference Between Echelon Form and Reduced Echelon Form

Echelon Form vs Reduced Echelon Form
 

The matrix obtained after performing several steps of the Gaussian elimination process is said to be in the echelon form or row-echelon form.

A matrix in the echelon form has the following properties.

• All the rows complete with zeros are at the bottom

• The first nonzero values in the nonzero rows shift to the right relative to the first nonzero term in the previous row (see example)

• Any nonzero row starts with 1

Following matrices are in the echelon form:

     

Continuing the elimination process gives a matrix with all the other terms of a column containing a 1 is zero. A matrix in that form is said to be in the reduced row echelon form.

       

But the above condition restricts the possibility of having columns with values except 1 and zero. For example, the following is also in the reduced row echelon form.

The reduced row echelon form is found when solving a linear system of equation using Gaussian elimination. The coefficient matrix of the matrix yields the reduced row echelon form and the solution/values for each individual can be easily obtained from a simple computation.

What is the difference between Echelon and Reduced Echelon Form?

• Row echelon form is one format of a matrix obtained by Gaussian elimination process.

• In Row echelon form, the non-zero elements are at the upper right corner, and every nonzero row has a 1. First nonzero element in the nonzero rows shifts to the right after each row.

• Further process of Gaussian elimination gives an even more simplified matrix, where all the other elements in a column containing 1 are zero. A matrix in that form is said to be in reduced row echelon form. That is, in reduced row echelon form, there can be no column that includes 1 and a value other than zero.