Skew Symmetric Matrix

Q.

For $3\times 3$ matrices M and N, which of the following sttements (s) is/are not correct?

1. $N^T$ MN is symmetric or skew – symmetric, accroding as M is symmetric or skew – symmetric.

2. MN – NM is symmetric for all symmetric matrices M and N

3. MN is symmetric for all symmetric matrices M and N

4. (adj M) (adj N) = adj(MN) for all invertible matrices M and N

Q.

Which of the following is true for a matrix A.

1. A = A^T⇒ A is skew symmetric

2. A = -A^T⇒A is skew symmetric

3. A = ⇒ A is skew symmetric

4. A = -A^T⇒ A is symmetric

Q.

Do symmetric matrices commute?

Q.

Is the transpose of a matrix symmetric?

Q.

$IfA=\left[\begin{array}{ccc}2& 1& 3\\ 1& 4& 5\\ 3& 5& 6\end{array}\right]B=\left[\begin{array}{ccc}1& 4& 5\\ 4& 2& 7\\ 5& 8& 3\end{array}\right]$ then which of the following is correct?

1. Both A and B are symmetric matrices

2. A is a symmetric matrix and B is not a symmetric matrix

3. B is a symmetric matrix and A is not a symmetric matrix

4. Both A and B are not symmetric matrices

Q. If U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A={1, 2, 3, 4, 5}, A symmetric difference B={1, 2, 3, 4} then find n[B-A].

Q. Let A and B are two non-singular square matrices, $A^T$ and $B^T$ are the transpose matrices of A and B respectively, then which of the following is correct

1. $B^{T}AB$ is symmetric matrix if and only if B is symmetric
2. $B^{T}AB$ is symmetric matrix if and only if A is symmetric
3. $B^{T}AB$ is skew symmetric matrix for every matrix A
4. $B^{T}AB$ is skew symmetric matrix if B is skew symmetric

Q. If A is a skew-symmetric matrix of order 3, then the matrix $A^4$ is

1. skew symmetric
2. diagonal
3. none of those
4. symmetric

Q.

If A and B are two square matrices which are skew symmetric then $(AB{)}^T$ equals to

1. BA
2. -AB
3. -BA
4. AB

Q. if A and B be two sets such that n(A)=15, n(B)=25, then number of possible values of symmetric difference of A and B is

Q. Choose the odd one number. Options: a)283 b)223 c)227 d)203

Q. Let X and Y be two arbitary, 3 $\times$ 3 non – zero, skew – symmetric matrices and Z be an arbitary, 3 $\times$ 3 non zero, symmetric matrix. Then, which of the following matrices is/are skew-symmetric?

1. $Y^3{Z}^4-Z^4{Y}^3$
2. $X^{44}+Y^{44}$
3. $X^4{Z}^3-Z^3{X}^4$
4. $X^{23}+Y^{23}$