Unit - 1

Linear Algebra- Matrices

1. Reduce the following matrix into normal form and find its rank,

Let A =

Apply we get

A 

Apply we get

A 

Apply

A 

Apply

A 

Apply 

A 

Hence the rank of matrix A is 2 i.e. .

2.       Reduce the following matrix into normal form and find its rank,

Let A =   

Apply and

A

Apply

A

Apply

A

Apply

A

Apply

A

Hence the rank of the matrix A is 2 i.e. .

3.       Reduce the following matrix into normal form and find its rank,

Let A =

Apply

Apply

Apply

Apply 

Apply and

Apply

Hence the rank of matrix A is 2 i.e. .

4.       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 coefficient matrix is equal to the no. Of unknowns. i.e. r = n = 3.

Hence the system has unique trivial solution.

i.e.

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

5.       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 coefficient matrix is non zero.

consider .

.

i.e.

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

6.       Determine the eigen values of eigen vector of the matrix.

Solution:

Consider the characteristic equation as,

i.e.

i.e.

i.e.

Which is the required characteristic equation.

are the required eigen values.

Now consider the equation

        … (1)

Case I:

If   Equation (1) becomes

R1 + R2

Thus

independent variable.

Now rewrite equation as,

Put x3 = t

&

Thus .

Is the eigen vector corresponding to .

Case II:

If equation (1) becomes,

Here

independent variables

Now rewrite the equations as,

Put

&

.

Is the eigen vector corresponding to .

Case III:

If equation (1) becomes,

Here rank of

independent variable.

Now rewrite the equations as,

Put

Thus .

Is the eigen vector for .

7.       Find the eigen values of eigen vector for the matrix.

Solution:

Consider the characteristic equation as

i.e.

i.e.

are the required eigen values.

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 eigen vector for

Case II:

If equation (1) becomes,

Thus

Independent variables.

Now rewrite the equations as,

Put

Is the eigen vector for

Now

Case II:-

If equation (1) gives,

R1 – R2

Thus

 independent variables

Now

Put

Thus

Is the eigen vector for .