RANK OF THE MATRIX USING ECHELON FORM

RANK OF THE MATRIX:
DEFINITION OF RANK : The rank of a matrix A is said to of rank 'r' ,

• if all of its (r+1) order minor A is zero
• there exists atleast one rth order minor of A which is not zero.

Rank of A is denoted by ρ(A).

NOTE:

1. Rank of a matrix is unique
2. Every matrix will have a rank
3. Rank of identity matrix of order nxn is 'n'
4. The rank of a matrix is >= r, if there is atleast one minor of rth order which is not equal to zero.
5. The rank of a matrix doesnt change if elementary row operations are performed.

ECHELON FORM OF MATRIX:

A matrix is said to be in echelon form if it has the following property

1. Zero rows are below the non zero rows
2. The number of zeros before the first nonzero element in a row is less than the no. of zeros in the next row.

The echelon form of the matrix can be obtained by performing ROW OPERATION only.

RANK OF THE MATRIX USING ECHELON FORM:

THE RANK OF THE MATRIX IS DETERMINED BY THE NO. OF NON ZERO ROWS IN THE ECHELON FORM OF THE MATRIX.

EXAMPLE FOR FINDING THE RANK

Find the rank of the matrix

1 -2 0 1

2 -1 1 0

3 -3 1 1

-1 -1 -1 1

Soln: R2 -> R2 - 2R1, R3 -> R3-3R1 and R4 -> R4+R1, we get

1 -2 0 1

0 3 1 -2

0 3 1 -2

0 -3 -1 2

R3-> R3-R2 and R4-> R4+R2

1 -2 0 1

0 3 1 -2

0 0 0 0

0 0 0 0

This matrix is in Echelon form. In the above matrix number of non-zero rows is 2.

Hence rank (A) = 2

IMPORTANT QUESTIONS :

Define the rank of the matrix and find the rank of the following matrix

2 1 3 5

4 2 1 3

8 4 7 13

8 4 -3 -1

Soln: the row operations to be followed are--

R2-> R2-2R1 ; R3->R3-4R1 ; R4->R4-4R1

Then R3-> R3- R2 ; R4-> R4-3R2

Ans: Rank of A = 2