Module 1

Matrices

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.      

 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.

5.     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.

6.     Verify Cayley – Hamilton theorem and use it to find A4 and A-1

Ex. Verify Cayley – Hamilton theorem and hence find A-1, A-2, A-3

Ex. For find the value of , using Cayley Hamilton theorem.

Ex. Find the characteristic equation of the matrix

And hence find the matrix represented by

Verify whether the following matrix is orthogonal or not if so find A-1

7.     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.

8.     Examine whether the following vectors are linearly independent or not.

and .

Solution:

Consider the vector equation,

i.e.   … (1)

Which can be written in matrix form as,

R12

R2 – 3R1, R3 – R1

R3 + R2

Here Rank of the coefficient matrix is equal to the no. of unknowns. i.e. r = n = 3.

Hence the system has a unique trivial solution.

i.e.

i.e. vector equation (1) has an only trivial solution. Hence the given vectors x1, x2, x3 are linearly independent.

9. At what value of P the following vectors are linearly independent.

Solution:

Consider the vector equation.

i.e.

Which is a homogeneous system of three equations in 3 unknowns and has a unique trivial solution.

If and only if  Determinant of the coefficient matrix is non zero.

consider .

.

i.e.

Thus for the system has an only trivial solution and Hence the vectors are linearly independent.

10.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 .

11. Find the eigen values 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 .