Let \(R=\left(\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)\) 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: \((I)\) \(Trace(\mathrm{R})=0\) \((II)\) If \(trace(\operatorname{adj}(\operatorname{adj}(\mathrm{R}))=0\), then \(R\) has exactly one non-zero entry.

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