Unit-5
Matrices
Question and Answer
- 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
- 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.
