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