If $AB = A$ and $BA = B$,then

If $A(\theta)=\begin{bmatrix} i \sin \theta & \cos \theta \\ \cos \theta & i \sin \theta \end{bmatrix}$ is a matrix,where $i=\sqrt{-1}$,then which of the following is not true?

Let $A$ be a $3 \times 3$ matrix such that $X^{T}AX = O$ for all nonzero $3 \times 1$ matrices $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $A \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}1 \\ 4 \\ -5\end{array}\right]$,$A \left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 4 \\ -8\end{array}\right]$,and $\operatorname{det}(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma$,where $\alpha, \beta, \gamma \in \mathbb{N}$,then $\alpha^2+\beta^2+\gamma^2$ is

Let $A = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix}$ where $i = \sqrt{-1}$. Then,the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$ is

Let $z = \frac{-1 + \sqrt{3}i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1, 2, 3\}$. Let $P = \begin{bmatrix} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{bmatrix}$ and $I$ be the identity matrix of order $2$. Then the total number of ordered pairs $(r, s)$ for which $P^2 = -I$ is

Let $P$ be a square matrix such that $P^2 = I - P$. For $\alpha, \beta, \gamma, \delta \in N$,if $P^\alpha + P^\beta = \gamma I - 29 P$ and $P^\alpha - P^\beta = \delta I - 13 P$,then $\alpha + \beta + \gamma - \delta$ is equal to $........$.