Reduced Row Echelon Form

A matrix A is said to be in Echelon form if either A is a null matrix or matrix A satisfies following conditions:
1.) Every non zero row in A precedes every zero row.
2.) Before the first non zero element in a row the number of zeros is less than the number of such zeros in the next row

The above matrix is in Row Echelon form but not in Row Reduced echelon form.
It can be easily proved that the rank of a matrix in Echelon form is equal to the number of non zero rows of the matrix.
Properties of a Matrix in reduced Row Echelon form
Following are the propreties of a matrix in reduced row echelon form.
1. The first entry is 1. The first entry is called a leading 1.The leading entry is also called Pivot.
2. All rows having elements as zero are at the bottom of the matrix.
3. All other elements in that column are 0 in which a column contains a leading 1.
4. The first non zero number occurs further to the right than the previous row as the rows are followed from top to bottom.

The above matrix is in Row reduced Echelon form.

The above Matrix is not in reduced row echelon form as according to the property of Reduced Row Echelon form the main diagonal is made up of 1and also the leading coefficient in row 3 is not to the right of leading coefficient of row 2.
To Row reduce a matrix we apply row operation on matrix that is we convert the matrix into a matrix where the first m x m entries form the identity matrix.
To reduce any matrix we can apply 3 operations on row.

• we can switch 2 rows.

• we can multiply a row by a constant

• we can add constant times row to another row.

When we get augmented matrix in row echelon form instead of stopping there we apply additional row transformations to transform the leading entries to 1 and to make out zero all entries above the leading Non zero entries. A matrix in this form is called Row Reduced Echelon form.