$\Rightarrow x, y, z \neq 0 \text { and } \sin \theta, \sin \left(\theta+\frac{2 \pi}{3}\right), \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0$

Also,$\sin \theta+\sin \left(\theta+\frac{2 \pi}{3}\right)+\sin \left(\theta+\frac{4 \pi}{3}\right)=0 \forall \theta \in \mathrm{R} $

$\Rightarrow \mathrm{x}+\mathrm{y}+\mathrm{z}=\frac{-\lambda}{2} \frac{\left(\sin ^2 \theta+\sin ^2\left(\theta+\frac{2 \pi}{3}\right)+\sin ^2\left(\theta+\frac{4 \pi}{3}\right)\right)}{\sin \theta \sin \left(\theta+\frac{2 \pi}{3}\right) \sin \left(\theta+\frac{4 \pi}{3}\right)} \neq 0$

$(i)$ $\quad$ Trace $(\mathrm{R})=x+y+z \neq 0$

$\Rightarrow$ Statement $(i)$ is False

$(ii)$ $\operatorname{Adj}(\operatorname{Adj}(\mathrm{R}))=|\mathrm{R}| \mathrm{R}$

Trace $(Adj(Adj(R)))$

$=x y z(x+y+z) \neq 0$

$\Rightarrow$ Hypothesis of conditional statement $(ii)$ is false

$\Rightarrow$ Conditional statement $(ii)$ is vacuously true !!