**rank of a matrix**is the dimension of its row space or column space. It is important to note that the row space and column space of a matrix have equal dimensions.

### Definition of rank of a matrix

A nimber r is said to be a rank of a non-zero matrix A if**1.**There exist at least one minor of order r of A which does not vanish and

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

The rank of a matrix A is denoted by .

We have

#### Another definition of rank of a matrix

The rank of a non-zero matrix is largest order of any non-vanishing minor of the matrix.**Remarks**

From the above definition of rank of a matrix, we observe that

**1.**The rank of zero matrix is zero i.e., where O is a zero matrix,

**2.**The rank of a non-singular matrix of order n is n,

**3.**, if every minor of order r+1 vanishes,

**4.**, if there is a minor of order r which does not vanish.

### ☆Finding rank of matrix

**1.**Minor method

**2.**Normal form

**3.**Echelon form of matrix

#### 1. Example on minor method

**Q1.**Determine the rank of matrix

**Ans**

A is non singular

**Q2.**Determine the rank of matrix

**Ans.**

i.e, only minor of order 3 of A vanishes.

Now we consider any minor of order 2.

Consider

There is a minor of order 2 of A which does not vanish

#### 2. Normal form of a matrix

are called the normal formes of matrix

#### Example

**Q3.**Prove that the matrix is equivalent to .

**Ans**

Given matrix is equivalent to .

#### 3. Echelon form

A matrix is said to be in echelon form if**1.**The zero rows of A occur below all the non-zero rows of A

**2.**The number of rows before the first non-zero element in a row is less than the number of such zero in the next row.

#### Example

**Q4.**Reduce to row echelon form the matrix

**Ans.**

Which is in row echelon form.

Since there are two non-zero rows in row echelon form.