JEE Math Practice Question

Let $$R=\left(\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$$ be a non-zero $$3 \times 3$$ matrix, where $$x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$$. For a square matrix $$M$$, let trace $$(M)$$ denote the sum of all the diagonal entries of $$M$$. Then, among the statements: (I) Trace $$(R)=0$$ (II) If trace $$(\operatorname{adj}(\operatorname{adj}(R))=0$$, then $$R$$ has exactly one non-zero entry.

1. A.Only (I) is true
2. B.Only (II) is true
3. C.Both (I) and (II) are true
4. D.Neither (I) nor (II) is true