Skew Symmetric Matrix

Q. If A and B are symmetric matrices then (AB + BA) is a matrix and (AB - BA) is a matrix.

1. skew - symmetric
2. symmetric
3. null
4. identity

Q.

All the diagonal entries of a skew symmetric matrix are positive.

1. True

2. False

Q.

If A is a square matric $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then A - $A^T$ is symmetric.

1. True

2. False

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. $X^{23}+Y^{23}$

Q.

Which of the following is true for a matrix A.

1. $A=A^T\Rightarrow$ A is skew symmetric

2. $A=-A^T\Rightarrow$ A is skew symmetric

3. $A=\overline{A}\Rightarrow$ A is skew symmetric

4. $A=-A^T\Rightarrow$A is symmetric

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. Let $X$ and $Y$ be two arbitrary, $3\times 3$, non-zero, skew symmetric matrices and $Z$ be an arbitrary $3\times 3$, 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. $X^{23}+Y^{23}$