**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?**

Sol

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.

**Int****erested in learning about similar topics? Here are a few hand-picked blogs for you!**