Unit 1

Unit 1

Matrices

Question and answer

1. Reduce the following matrix to the normal form of Hence find it’s rank,

Solution:

We have,

Apply

The rank of A = 1

2.     Find the rank of the matrix

Solution:

We have,

Apply R12

The rank of A = 3

3.     Find the rank of the following matrices by reducing it to the normal form.

Solution:

Apply C14

4.     Reduce the following matrix to the normal form of Hence find it’s rank,

Solution:

We have,

Apply

The rank of A = 1

5.     If Find Two

Matrices P and Q such that PAQ is in normal form.

Solution:

Here A is a square matrix of order 3 x 3. Hence, we write,

A = I3 A.I3

i.e.

i.e.

6.     Are the vectors , , linearly dependent. If so, express x1 as a linear combination of the others.

Solution:

Consider a vector equation,

i.e.

Which can be written in matrix form as,

Here & no. Of unknown 3. Hence the system has infinite solutions. Now rewrite the questions as,

Put

and

Thus

i.e.

i.e.

Since F11 k2, k3 not all zero. Hence are linearly dependent.

7.     Determine the eigen values of the eigenvector of the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

i.e.

Which is the required characteristic equation.

are the required eigenvalues.

Now consider the equation

… (1)

Case I:

If Equation (1)becomes

R1 + R2

Thus

independent variable.

Now rewrite the equation as,

Put x3 = t

&

Thus .

Is the eigenvector corresponding to .

Case II:

If equation (1) becomes,

Here

independent variables

Now rewrite the equations as,

Put

&

.

Is the eigenvector corresponding to .

Case III:

If equation (1) becomes,

Here the rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigenvector for .

8.     Find the eigenvalues of an eigenvector for the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

are the required eigenvalues.

Now consider the equation

… (1)

Case I:

Equation (1) becomes,

Thus and n = 3

3 – 2 = 1 independent variables.

Now rewrite the equations as,

Put

,

i.e. the eigenvector for

Case II:

If equation (1) becomes,

Thus

Independent variables.

Now rewrite the equations as,

Put

Is the eigenvector for

Now

Case II:-

If equation (1) gives,

R1 – R2

Thus

independent variables

Now

Put

Thus

Is the eigenvector for .

9.     Find a non – singular matrices p and Q such that P A Q is in a normal form where

Solution:

Here A is a matrix of order 3 x 4. Hence, we write A as,

i.e.

i.e.

Unit 1

Matrices

Question and answer

1. Reduce the following matrix to the normal form of Hence find it’s rank,

Solution:

We have,

Apply

The rank of A = 1

2.     Find the rank of the matrix

Solution:

We have,

Apply R12

The rank of A = 3

3.     Find the rank of the following matrices by reducing it to the normal form.

Solution:

Apply C14

4.     Reduce the following matrix to the normal form of Hence find it’s rank,

Solution:

We have,

Apply

The rank of A = 1

5.     If Find Two

Matrices P and Q such that PAQ is in normal form.

Solution:

Here A is a square matrix of order 3 x 3. Hence, we write,

A = I3 A.I3

i.e.

i.e.

6.     Are the vectors , , linearly dependent. If so, express x1 as a linear combination of the others.

Solution:

Consider a vector equation,

i.e.

Which can be written in matrix form as,

Here & no. Of unknown 3. Hence the system has infinite solutions. Now rewrite the questions as,

Put

and

Thus

i.e.

i.e.

Since F11 k2, k3 not all zero. Hence are linearly dependent.

7.     Determine the eigen values of the eigenvector of the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

i.e.

Which is the required characteristic equation.

are the required eigenvalues.

Now consider the equation

… (1)

Case I:

If Equation (1)becomes

R1 + R2

Thus

independent variable.

Now rewrite the equation as,

Put x3 = t

&

Thus .

Is the eigenvector corresponding to .

Case II:

If equation (1) becomes,

Here

independent variables

Now rewrite the equations as,

Put

&

.

Is the eigenvector corresponding to .

Case III:

If equation (1) becomes,

Here the rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigenvector for .

8.     Find the eigenvalues of an eigenvector for the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

are the required eigenvalues.

Now consider the equation

… (1)

Case I:

Equation (1) becomes,

Thus and n = 3

3 – 2 = 1 independent variables.

Now rewrite the equations as,

Put

,

i.e. the eigenvector for

Case II:

If equation (1) becomes,

Thus

Independent variables.

Now rewrite the equations as,

Put

Is the eigenvector for

Now

Case II:-

If equation (1) gives,

R1 – R2

Thus

independent variables

Now

Put

Thus

Is the eigenvector for .

9.     Find a non – singular matrices p and Q such that P A Q is in a normal form where

Solution:

Here A is a matrix of order 3 x 4. Hence, we write A as,

i.e.

i.e.