Skew Hermitian Matrix

Q. If $M$ is any square matrix of order $3$ over $R$ and if $M^{\prime }$ be the transpose of $M$, then $\text{adj}\left(M^{\prime }\right)-\text{adj}(M{)}^{\prime }$ is equal to

1. $M$
2. $M^{\prime }$
3. null matrix
4. identity matrix

Q.

Which among the following is/are skew hermitian matrix.

1. 

2. 

3. 

4. 

Q.

If A is a square matrix then $A-\overline{A^T}$ is a skew hermitian matrix.

1. False

2. True

Q.

What is the necessary condition for a matrix A to be invertible?

1. |A|=|A^T|

2. |A|≠0

3. A ≠ 0

4. |A|=|Adj A|

Q.

What is the element in the 2^nd row and 1^stcolumn of a 2 x 2 Matrix A= [ a_ij], such that a ₌ (i + 3) (j - 1)

1. -5

2. 5

3. 4

4. 0

Q.

1.If A and B are non-singular matrices, then

[MP PET 1991; Kurukshetra CEE 1998]

1. 
   

2. 
   

3. 
   

4. 
   

Q.

In a square matrix of order 4, the element $a_{3}3=33$. This square matrix cannot be skew hermitian.

1. true

2. False

Q.

Which of the following represents the condition for a matrix A to be skew hermitian Matrix. Given that the general element of the matrix is aij.

1. 

2. 

3. 

4. 

Q. If $0$ is a characteristic root of $A$, then :

1. $A=0$
2. $A=I_n$
3. $A$ is non-singular
4. $A$ is singular

Q. If the product of two non-null square matrix is a null matrix, then

1. only one of them is singular
2. both of them must be singular
3. both of them are non singular
4. none of these

Q.

Which of the following represents the condition for a matrix A to be skew hermitian Matrix. Given that the general element of the matrix is aij.

1. $aij=-\overline{aij}$

2. $aij=\overline{aji}$

3. $aij=-\overline{aji}$

4. $\overline{aij}=-aij$

Q. Let two matrices $A=\text{diag}(-1,\omega ,i)$ and $B=\text{diag}(1,{\omega }^2,-i),$ then which of the following statement(s) is(are) correct ?
$(\text{where}{i}^2=-1\text{and}\omega \text{is a cube root of unity.})$

1. $Tr\left(AB\right)=\omega$
2. $Tr\left(A^2\right)={\omega }^2$
3. $Tr\left(B^2\right)=\omega$
4. $Tr\left(A^2{B}^2\right)=3$

Q.

If A is a square matrix then $A-\overline{A^T}$ is a skew hermitian matrix.

1. True

2. False

Q. Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M → R defined by f(A) = |A| for every A ∈ M, is
(a) one-one and onto
(b) neither one-one nor onto
(c) one-one but-not onto
(d) onto but not one-one

Q.

If A is a square matrix then $A-\overline{A^T}$ is a skew hermitian matrix.

1. True

2. False

Q.

In a square matrix of order 4, the element $a_{33}$ = 33. This square matrix cannot be skew hermitian.

1. True

2. False

Q. If the matrices A, B and A+B are invertible then [A(A+B)^-1 B]^-1 is equal to