Unit-5

Matrices

Question and Answer

1. Let A =  Find .

Then

2.     let A =  Find A-1

Then

3.     Express   the matrix     as sum of  a symmetric and  skew  symmetric matrix.

Let    A =   therefore A’ =

Suppose

We calculate  

Hence P is the required symmetric matrix.

Again   let 

We find 

Hence Q is the required skew -symmetric matrix.

Now,   we check

This implies that square matrix   A can be   represented as   sum of symmetric and skew symmetric matrices.

4.     Show that     is a   Hermitian matrix.

Let  A = therefore A’ = 

  Now,

Here 

Hence given matrix is a Hermitian matrix

5.     Show that

Let A =   and then A’=  

It is clear, hence the given matrix is skew-hermitian   matrix.

6.     Express the matrix A as sum of hermitian and skew-hermitian matrix where

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.

7.     Prove that the following matrix is orthogonal:

Let A  =   then  A’= 

Now,   AA’=

   Thus AA’=   = I

Hence the matrix is orthogonal.

8.     Prove that

Let U = therefore   U’= 

Also

Now,  UU’=

Hence the given matrix is a unitary matrix.

9.     Find the rank of the following matrices?

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 

10. Reduce the matrix into normal form and find the rank of matrix:

Let A =

Apply

Apply

Apply

Apply 

Apply  

Apply 

Apply 

Hence the rank of the matrix is 3.

11. Find the non-singular matrices P and Q such that PAQ is in normal form:

Let A =  

Also we know that  A  =A where all row elementary  transformation is  applied in identity matrix pre multiplied where  as column elementary transformation  is  applied in identity  matrix post  multiplied.

                A

              Apply

               A

          Apply

               A

         Apply

         A

       Apply  

            A

        Apply 

           A

     Or       where P= , Q=

Hence the rank of matrix A is 2.

Unit-5

Matrices

Question and Answer

1. Let A =  Find .

Then

2.     let A =  Find A-1

Then

3.     Express   the matrix     as sum of  a symmetric and  skew  symmetric matrix.

Let    A =   therefore A’ =

Suppose

We calculate  

Hence P is the required symmetric matrix.

Again   let 

We find 

Hence Q is the required skew -symmetric matrix.

Now,   we check

This implies that square matrix   A can be   represented as   sum of symmetric and skew symmetric matrices.

4.     Show that     is a   Hermitian matrix.

Let  A = therefore A’ = 

  Now,

Here 

Hence given matrix is a Hermitian matrix

5.     Show that

Let A =   and then A’=  

It is clear, hence the given matrix is skew-hermitian   matrix.

6.     Express the matrix A as sum of hermitian and skew-hermitian matrix where

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.

7.     Prove that the following matrix is orthogonal:

Let A  =   then  A’= 

Now,   AA’=

   Thus AA’=   = I

Hence the matrix is orthogonal.

8.     Prove that

Let U = therefore   U’= 

Also

Now,  UU’=

Hence the given matrix is a unitary matrix.

9.     Find the rank of the following matrices?

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 

10. Reduce the matrix into normal form and find the rank of matrix:

Let A =

Apply

Apply

Apply

Apply 

Apply  

Apply 

Apply 

Hence the rank of the matrix is 3.

11. Find the non-singular matrices P and Q such that PAQ is in normal form:

Let A =  

Also we know that  A  =A where all row elementary  transformation is  applied in identity matrix pre multiplied where  as column elementary transformation  is  applied in identity  matrix post  multiplied.

                A

              Apply

               A

          Apply

               A

         Apply

         A

       Apply  

            A

        Apply 

           A

     Or       where P= , Q=

Hence the rank of matrix A is 2.