Let R= (x, 0, 0; 0, y, 0; 0, 0, z) be a non-zero 3 × 3 matrix, where x sin θ=y sin (θ+(2 π)/3)=z sin (θ+(4 π)/3) ≠ 0, θ ∈(0,2 π). For a square matrix…

Let $\displaystyle R=\left(\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $\displaystyle 3 \times 3$ matrix, where $\displaystyle x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0, \theta \in(0,2 \pi)$. For a square matrix $\displaystyle M$, let trace ( $\displaystyle M$ ) denote the sum of all the diagonal entries of $\displaystyle M$. Then, among the statements:

$\displaystyle \operatorname{Trace}(R)=0$

If $\displaystyle \operatorname{trace}(\operatorname{adj}(\operatorname{adj}(R))=0$, then $\displaystyle R$ has exactly one non-zero entry.