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Unit-4Linear algebraQ1) Find the inverse of matrix ‘A’ by using elementary transformation-

A =

A1)

Write the matrix ‘A’ as-  

A = IA

Apply , we get

Apply

Apply

Apply

Apply

So that,

  =

 Q2) Find the rank of a matrix M by echelon form.

M = 

A2)

First we will convert the matrix M into echelon form,

M =

Apply, , we get

M =

Apply  , we get

M =

 

Apply 

M =

We can see that, in this echelon form of matrix, the number of non – zero rows is 3.

So that the rank of matrix X will be 3.

Q3) Find the rank of a matrix A by echelon form.

A =

A3)

Transform the matrix A into echelon form, then find the rank,

We have,

A =

Apply, 

A =

Apply ,

A =

Apply

A =

Apply

A =

Hence the rank of the matrix will be 2.

 Q4) reduce the matrix A to its normal form and find rank as well.

A4)

We have,

We will apply elementary row operation,

We get,

Now apply column transformation,

We get,

Apply

  , we get,

Apply   and

Apply 

Apply   and

Apply   and

As we can see this is required normal form of matrix A.

Therefore, the rank of matrix A is 3.

 Q5) Find out the value of ‘b’ in the system of homogenenous equations-

2x + y + 2z = 0

x + y + 3z = 0

4x + 3y +  bz = 0

Which has

(1) Trivial solution

(2) Non-trivial solution

A5)

For trivial solution, we already know that the values of x , y and z will be zerp, so that ‘b’ can have any value.

Now for non-trivial solution-

(2)

Convert the system of equations into matrix form-

AX = O

Apply Respectively , we get the following resultant matrices

 

For non-trivial solutions, r(A) = 2 < n

b – 8 = 0

b = 8

 Q6) Check the consistency and find the values of x , y and z of the following system of linear equations.2x + 3y + 4z = 11X + 5y + 7z = 153x + 11y + 13z = 25A6)

Re-write the system of equations in augmented matrix form.

                                  C = [A,B] 

That will be,

Apply 

Now apply ,

We get,

~~

Here rank of A = 3

And rank of C = 3, so that the system of equations is consistent,

So that we can can solve the equations as below,

That gives,

x + 5y + 7z = 15  …………..(1)

y + 10z/7 = 19/7 ………………(2)

4z/7 = 16/7 ………………….(3)

From eq. (3)

z = 4,

From 2,

From eq.(1), we get

x + 5(-3) + 7(4) = 15

That gives,

x = 2

Therefore the values of x , y , z are    2 , -3 , 4  respectively.

Q6) Solve the following equations by using Cramer’s rule-

A6)

Here we have-

And here-

Now by using cramer’s rule-

Q7) solve the following system of linear equations by using Guassseidel method-6x + y + z = 1054x + 8y + 3z = 1555x + 4y  - 10z = 65A7)

The above equations can be written as,

   ………………(1)

………………………(2)

   ………………………..(3)

Now put z = y = 0 in first eq.

We get

x = 35/2

put x = 35/2 and z = 0 in eq. (2)

we have,

Put the values of x and y in eq. 3

Again start from eq.(1)

By putting the values of y and z

y = 85/8 and z = 13/2

We get

The process can be showed in the table format as below

At the fourth iteration , we get the values of x = 14.98 , y = 9.98  , z = 4.98

Which are approximately equal to the actual values,

As x = 15  , y = 10 and  y = 5 ( which are the actual values)

 Q8) Show that,

A8)

Applying

          We get,

 

Example: Solve-

Sol:

given-

Apply-

We get-

 Q9)

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

A9)

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 F11k2, k3 not all zero. Hence are linearly dependent.

 Q10) Diagonalise the matrix A10)

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    

 

 Q11) Find the characteristic equation of the the matrix A and verify Cayley-Hamilton theorem as well.

A =

 A11)

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.

 Q12) Verify the Cayley-Hamilton theorem and find the inverse.

?

A12)

Let A =

The characteristics equation of A is

Or

Or 

Or 

By Cayley-Hamilton theorem 

                                   L.H.S:

                                                 =   =0=R.H.S

Multiply both side by on 

                                               Or

                                              Or [

                                               Or