Let $\mathrm{x} \in \mathbb{R}$ and let $\mathrm{P}=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \mathrm{Q}=\left[\begin{array}{lll}2 & \mathrm{x} & \mathrm{x} \\ 0 & 4 & 0 \\ \mathrm{x} & \mathrm{x} & 6\end{array}\right]$ and $\mathrm{R}=P Q P^{-1}$. Then which of the following options is/are correct?
Select ALL correct options:
A
For $\mathbf{x}=1$, there exists a unit vector $\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ for which $\mathbf{R}\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$
B
There exists a real number x such that PQ = QP
C
$\operatorname{det} R=\operatorname{det}\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right]+8$, for all $x \in \mathbb{R}$
D
For $x=0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]=6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a+b=5$
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