Question Bank ( unit-2)

Question Bank ( unit-2)

Question-1: find the linear transformation of the matrix A.

A =

Sol. We have,

A =

Multiply the matrix by vector x = (x , y , z) , we get

Ax =

= ( )

f(x , y , z) = ( )

Which is the linear transformation of A.

Question-2: 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.

Question-3: At what value of P the following vectors are linearly independent.

Solution:

Consider the vector equation.

i.e.

This 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.

Question-4: 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 .

Question-5: : Diagonalise the matrix

Sol.

Let A=

The three Eigen vectors obtained are (-1,1,0), (-1,0,1) and (3,3,3) corresponding to Eigen values .

Then and

Also we know that

Question-6: Diagonalise the matrix

Sol.

Let A =

The Eigen vectors are (4,1),(1,-1) corresponding to Eigen values .

Then and also

Also we know that

Question-7: Find the characteristic equation of the matrix A = and Verify cayley-Hamlton theorem.

Sol. Characteristic equation of the matrix, we can be find as follows-

Which is,

( 2 - , which gives

According to cayley-Hamilton theorem,

…………(1)

Now we will verify equation (1),

Put the required values in equation (1) , we get

Hence the cayley-Hamilton theorem is verified.

Question-8: Find the characteristic equation of the the matrix A and verify Cayley-Hamilton theorem as well.

A =

Sol. Characteristic equation will be-

= 0

( 7 -

(7-

(7-

Which gives,

Or

According to cayley-Hamilton theorem,

…………………….(1)

In order to verify cayley-Hamilton theorem , we will find the values of

So that,

Now

Put these values in equation(1), we get

= 0

Hence the cayley-hamilton theorem is verified.

Question-9: Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

Sol. The characteristic equation will be,

|A - | = 0

Which gives,

(4-

According to Cayley-Hamilton theorem,

Multiplying by

That means

On solving ,

11

=

=

So that,

Question-10: Find the inverse of matrix A by using Cayley-Hamilton theorem.

A =

Sol. The characteristic equation will be,

|A - | = 0

=

= (2-

= (2 -

=

That is,

Or

We know that by Cayley-Hamilton theorem,

…………………….(1)t,

Multiply equation(1) by , we get

Or

Now we will find

=

=

Hence the inverse of matrix A is,

Question-11: : Find of matrix A by using Cayley-Hamilton theorem.

Sol. First we will find out the characteristic equation of matrix A,

|A - | = 0

We get,

Which gives,

(

We get,

Or I ……………………..(1)

In order to find find we take cube of eq. (1)

We get,

729I we know that-

729 we know that- value of I =

Question-12: find out the quadratic form of following matrix.

A =

Solution: Quadratic form is,

X’ AX

Which is the quadratic form of a matrix.

Question-13: find the real matrix of the following quadratic form:

Sol. Here we will compare the coefficients with the standard quadratic equation,

We get,

Question-14: Find the orthogonal canonical form of the quadratic form.

5

Sol. The matrix form of this quadratic equation can be written as,

A =

We can find the eigen values of A as –

|A - | = 0

= 0

Which gives,

The required orthogonal canonical reduction will be,

8 .