$P$ is an orthogonal matrix and $A$ is a periodic matrix with period $4$,and $Q = PAP^T$. Then $X = P^TQ^{2005}P$ will be equal to

If $A$ is a symmetric matrix and $n \in N$,then $A^n$ is

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$,where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix,then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that

If the matrix $\begin{bmatrix} a & 2 & -3 \\ b & 0 & 4 \\ c & -4 & 0 \end{bmatrix}$ is a skew-symmetric matrix,then $a+b+c=$

If $ A=\begin{bmatrix} \cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{bmatrix} $ and $ A+A^{T}=I $,where $ I $ is the $ 2 \times 2 $ identity matrix and $ A^{T} $ is the transpose of $ A $,then the value of $ \theta $ is equal to

For a given matrix $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$,which of the following statements holds true?