Question Bank

Question-1: check whether the following matrix A is symmetric or not?

                                 A =      

Sol. As we know that if the transpose of the given matrix is same as the matrix itself then the matrix is called symmetric matrix.

So that, first we will find its transpose,

Transpose of matrix A ,

Here,

                                   A =

So that, the matrix A is symmetric.

Question-2: : Express the matrix A as sum of hermitian and skew-hermitian matrix where

Sol.

Let A =

Therefore    and

        Let

Again

Hence P is a hermitian matrix.

Let 

Again

Hence Q is a skew- hermitian matrix.

We Check

  P +Q=

Hence proved.

Question-3: If A =   then show that

(i) is hermitian matrix.

(ii)   is  skew-hermitian matrix.

Sol.

Given   A =

Then 

Let

Also  

Hence P is a Hermitian matrix.

Let

Also

Hence Q is a   skew-hermitian matrix.

Question-4: Find the rank of a matrix M by echelon form.

                               M = 

Sol. 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.

Question-5: Find the rank of a matrix A by echelon form.

                       A =

Sol. Convert the matrix A into echelon form,

                                 A =

Apply 

                                 A =

Apply    , we get

                                     A =

Apply , we get

                                   A =

Apply ,

                                   A =

Apply  ,

                                 A =

Therefore the rank of the matrix will be 2.

Question-6: :  Find the rank of a matrix A by echelon form.

                                 A =

Sol. 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.

Question-7: Find the rank of the following matrix by echelon form?

Sol.

Let A =

Applying

     A 

Applying

   A 

Applying

A 

      Applying

             A 

It is  clear  that  minor  of order 3  vanishes but  minor  of order  2  exists  as 

Hence rank of a given matrix A is 2 denoted by 

Question-8: reduce the matrix A to its normal form and find rank as well.

Sol.  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.

Question-9: Find the solution of the following homogeneous system of linear equations,

Sol. The given system of linear equations can be written in the form of matrix as follows,

Apply the elementary row transformation,

  , we get,

    , we get

  Here r(A) = 4, so that it has trivial solution,

Question-10: check whether the following system of linear equations is consistent of not.

2x + 6y = -11

6x + 20y – 6z = -3

6y – 18z = -1

Sol. Write the above system of linear equations in augmented matrix form,

Apply  , we get

Apply

Here the rank of C is 3 and the rank of A is 2

Therefore both ranks are not equal. So that the given system of linear equations is not consistent.

Question-11: Check the consistency and find the values of x , y and z of the following system of linear equations.

2x + 3y + 4z = 11

X + 5y + 7z = 15

3x + 11y + 13z = 25

Sol. 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.

Question-12: solve the following system of linear equations by using Guass seidel method-

5x + 2y  + z = 12

x + 4y + 2z = 15

x + 2y + 5z = 20

Sol. These equations can be written as ,

   ………………(1)

………………………(2)

   ………………………..(3)

Put y and z equals to 0 in eq. 1

We get,

x = 2.4

Put x = 2.4, and z = 0 in eq. 2 , we get

Put x = 2.4 and y = 3.15 in eq.(3) , we get

Again start from eq.(1), put the values of y and z , we get

                               = 0.688

We repeat the process again and again ,

The following table can be obtained –

We see that the values are apprx. Equal to exact values.

Exact values are , x = 1  , y = 2,   z = 3.

                                          Question Bank

Question-1: check whether the following matrix A is symmetric or not?

                                 A =      

Sol. As we know that if the transpose of the given matrix is same as the matrix itself then the matrix is called symmetric matrix.

So that, first we will find its transpose,

Transpose of matrix A ,

Here,

                                   A =

So that, the matrix A is symmetric.

Question-2: : Express the matrix A as sum of hermitian and skew-hermitian matrix where

Sol.

Let A =

Therefore    and

        Let

Again

Hence P is a hermitian matrix.

Let 

Again

Hence Q is a skew- hermitian matrix.

We Check

  P +Q=

Hence proved.

Question-3: If A =   then show that

(i) is hermitian matrix.

(ii)   is  skew-hermitian matrix.

Sol.

Given   A =

Then 

Let

Also  

Hence P is a Hermitian matrix.

Let

Also

Hence Q is a   skew-hermitian matrix.

Question-4: Find the rank of a matrix M by echelon form.

                               M = 

Sol. 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.

Question-5: Find the rank of a matrix A by echelon form.

                       A =

Sol. Convert the matrix A into echelon form,

                                 A =

Apply 

                                 A =

Apply    , we get

                                     A =

Apply , we get

                                   A =

Apply ,

                                   A =

Apply  ,

                                 A =

Therefore the rank of the matrix will be 2.

Question-6: :  Find the rank of a matrix A by echelon form.

                                 A =

Sol. 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.

Question-7: Find the rank of the following matrix by echelon form?

Sol.

Let A =

Applying

     A 

Applying

   A 

Applying

A 

      Applying

             A 

It is  clear  that  minor  of order 3  vanishes but  minor  of order  2  exists  as 

Hence rank of a given matrix A is 2 denoted by 

Question-8: reduce the matrix A to its normal form and find rank as well.

Sol.  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.

Question-9: Find the solution of the following homogeneous system of linear equations,

Sol. The given system of linear equations can be written in the form of matrix as follows,

Apply the elementary row transformation,

  , we get,

    , we get

  Here r(A) = 4, so that it has trivial solution,

Question-10: check whether the following system of linear equations is consistent of not.

2x + 6y = -11

6x + 20y – 6z = -3

6y – 18z = -1

Sol. Write the above system of linear equations in augmented matrix form,

Apply  , we get

Apply

Here the rank of C is 3 and the rank of A is 2

Therefore both ranks are not equal. So that the given system of linear equations is not consistent.

Question-11: Check the consistency and find the values of x , y and z of the following system of linear equations.

2x + 3y + 4z = 11

X + 5y + 7z = 15

3x + 11y + 13z = 25

Sol. 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.

Question-12: solve the following system of linear equations by using Guass seidel method-

5x + 2y  + z = 12

x + 4y + 2z = 15

x + 2y + 5z = 20

Sol. These equations can be written as ,

   ………………(1)

………………………(2)

   ………………………..(3)

Put y and z equals to 0 in eq. 1

We get,

x = 2.4

Put x = 2.4, and z = 0 in eq. 2 , we get

Put x = 2.4 and y = 3.15 in eq.(3) , we get

Again start from eq.(1), put the values of y and z , we get

                               = 0.688

We repeat the process again and again ,

The following table can be obtained –

We see that the values are apprx. Equal to exact values.

Exact values are , x = 1  , y = 2,   z = 3.