Elementary Transformations of a Matrix: Solved Example Problems

Example 1.13

Reduce the matrix  to a row-echelon form.

Solution

Note

This is also a row-echelon form of the given matrix.

So, a row-echelon form of a matrix is not necessarily unique.

Example 1.14

Reduce the matrix  to a row-echelon form.

Solution

Rank of a Matrix

Example 1.15

Find the rank of each of the following matrices:

Solution

(i) Let A =. Then A is a matrix of order 3× 3. So ρ(A) ≤ min {3, 3}  = 3. The highest  order of minors of A is 3 . There is only one third order minor of A .

 It is  =  3 (6− 6) − 2 (6−6) +  5 (3 − 3) = 0. So, ρ(A) < 3.

 Next consider the second-order minors of A .

We find that the second order minor  = 3 − 2 = 1 ≠ 0 . So ρ(A) = 2 .

(ii) Let A = . Then A is a matrix of order 3×4 . So ρ(A) ≤ min {3, 4}  = 3.

The highest order of minors of A is 3 . We search for a non-zero third-order minor of A . But

we find that all of them vanish. In fact, we have

 So, ρ(A) < 3. Next, we search for a non-zero second-order minor of A .

We find that  = -4+9 =5 ≠ 0 . So, ρ(A) = 2 .

Remark

Finding the rank of a matrix by searching a highest order non-vanishing minor is quite tedious when the order of the matrix is quite large. There is another easy method for finding the rank of a matrix even if the order of the matrix is quite high. This method is by computing the rank of an equivalent row-echelon form of the matrix. If a matrix is in row-echelon form, then all entries below the leading diagonal (it is the line joining the positions of the diagonal elements a11 , a22 , a33 ,L. of the matrix) are zeros. So, checking whether a minor is zero or not, is quite simple.

Example 1.16

Find the rank of the following matrices which are in row-echelon form :

Solution

(i) Let A = . Then A is a matrix of order 3 3 × and ρ(A) ≤ 3

 The third order minor |A| =   = (2) (3)( 1) = 6 ≠ 0 . So, ρ(A) = 3 .

Example 1.17

Find the rank of the matrix  by reducing it to a row-echelon form.

Solution

Let A = . Applying elementary row operations, we get

The last equivalent matrix is in row-echelon form. It has two non-zero rows. So, ρ (A)= 2.

Example 1.18

Find the rank of the matrix  by reducing it to a row-echelon form.

Solution

Let A be the matrix. Performing elementary row operations, we get

The last equivalent matrix is in row-echelon form. It has three non-zero rows. So, ρ(A) = 3 .

Elementary row operations on a matrix can be performed by pre-multiplying the given matrix by a special class of matrices called elementary matrices.