Mathematics Question from JEE Main 2024 · 30 January, Shift 2

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