Symmetric Matrix

Q. does an infinite set have infinite subsets

Q. If A is a square matrix $A+A^T$ is symmetric matrix, then
$A-A^T=$

1. Unit matrix
2. Symmetric matrix
3. Skew symmetric matrix
4. Zero matrix

Q. If a square matrix A is such that $A{A}^T=I=A^{T}A$, then$|A|$is equal to

1. $0$
2. $\pm 1$
3. $\pm 2$
4. None of these

Q. Square matrix of order N with $\left|A\right|=9$ and $\left|adj\left(adj\left(adjA\right)\right)\right|=3^{16}$ then number of elements in A is

1. $16$
2. $9$
3. $4$
4. $None$ $of$ $these$

Q. If $\omega \ne 1$ is the complex cube root of unity and matrix $H=\left[\begin{array}{cc}\omega & 0\\ 0& \omega \end{array}\right]$, then $H^{70}$ is equal to

1. $H$
2. $0$
3. $-H$
4. $H^2$

Q. If A is a square matrix $A+A^T$ is symmetric matrix, then
$A-A^T=$

1. Unit matrix
2. Symmetric matrix
3. Skew symmetric matrix
4. Zero matrix

Q.

A square matrix A is said to be a symmetric matrix if

1. A = -A^T

2. A = A^T

3. A =

4. A =

Q.

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

1. True

2. False

Q. If A and B are two non singular matrices and both are symmetric and commute each other then

1. $A^{-1}B$ is symmetric but $A^{-1}{B}^{-1}$ is not symmetric
2. $A^{-1}{B}^{-1}$ is symmetric but $A^{-1}B$ is not symmetric
3. Neither $A^{-1}B$ nor $A^{-1}{B}^{-1}$ are symmetric
4. Both $A^{-1}B$ and $A^{-1}{B}^{-1}$ are symmetric

Q. Out of the following which is a skew- symmetric matrix

1. $\left[\begin{array}{ccc}0& 4& 5\\ -4& 0& -6\\ -5& 6& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 1& -6\\ -5& 6& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}1& 4& 5\\ -4& 2& -6\\ -5& 6& 3\end{array}\right]$
4. $\left[\begin{array}{ccc}i+1& 4& 5\\ -4& i& -6\\ -5& 6& i\end{array}\right]$