Let R=((|gathered x 0 0 0 y 0 0 0 z gathered))| be a non-zero 3 3 matrix, where x =y ( +(2…

Question

Let $R=((|\begin{gathered} x \quad  0 \quad  0 \ 0 \quad  y \quad  0 \ 0 \quad  0 \quad  z \end{gathered}))|$ be a non-zero $3	imes 3$ matrix, where $x\sin 	heta =y\sin (	heta +\frac{2\pi }{3})=z\sin (	heta +\frac{4\pi }{3})
e 0,	heta \in (0,2\pi )$. 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 $(adj(adj(R))=0$, then $R$ has exactly one non-zero entry.

Accepted Answer

$$\begin{aligned}
& x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0 \
& \Rightarrow x, y, z \neq 0
\end{aligned}$$

Also,

$$\begin{aligned}
& \sin \theta+\sin \left(\theta+\frac{2 \pi}{3}\right)+\sin \left(\theta+\frac{4 \pi}{3}\right)=0 \forall \theta \in \mathrm{R} \
& \Rightarrow \frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}+\frac{1}{\mathrm{z}}=0 \
& \Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=0
\end{aligned}$$

(i) $$\quad \operatorname{Trace}(\mathrm{R})=\mathrm{x}+\mathrm{y}+\mathrm{z}$$

If $x+y+z=0$ and $x y+y z+z x=0$

$$\Rightarrow \mathrm{x}=\mathrm{y}=\mathrm{z}=0$$

Statement (i) is False

(ii) $$\quad \operatorname{Adj}(\operatorname{Adj}(\mathrm{R}))=|\mathrm{R}| \mathrm{R}$$

Trace $$(\operatorname{Adj}(\operatorname{Adj}(\mathrm{R})))$$

$$=x y z(x+y+z) \neq 0$$

Statement (ii) is also False

Solutions · 1