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What is the echelon form of a matrix?

by krishna

Overview(echelon form)

There are two types of echelon form of a matrix.

Row echelon form

A matrix satisfying the following conditions is said to be in the row echelon form-

Condition-1:  The first non-zero element (leading element) in each row should be 1.

Condition-2: Each leading element is in a column to the right of the leading element in the previous row.

Condition-3: The rows with all zero elements (if any) are at the bottom.

Inverse of a matrix by using elementary transformation

The following transformation, we call as elementary transformations-

1. Interchange of any two rows (column)

2. Multiplication of any row or column by any non-zero scalar quantity k.

3. Addition to one row (column) of another row(column) multiplied by any non-zero scalar.

The symbol ~ is used for equivalence.

Elementary matrices

If we get a square matrix from an identity or unit matrix by using any single elementary transformation is called elementary matrix.

Note- Every elementary row transformation of a matrix can be affected by pre multiplication with the corresponding elementary matrix.

Rank of a matrix by echelon form

The rank of a matrix (r) is, when–

1. It has at least one non-zero minor of order r.

2. Every minor of A of order higher than r is zero.

Solved examples

Example: Find the rank of a matrix M.

Sol. First we will convert the matrix M into echelon form,

We can see that, in this echelon form of matrix, the number of non – zero rows is 3.

So that the rank of matrix X will be 3.

Example: Find the rank of the following matrices by echelon form?


It is  clear  that  minor  of order 3  vanishes but  minor  of order  2  exists  as  

Hence rank of a given matrix A is 2.

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