Matrices And Determinants

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.