Matrices Question 8
Question 8 - 2024 (30 Jan Shift 2)
Let $R=\left(\begin{array}{lll}x & 0 & 0 \ 0 & y & 0 \ 0 & 0 & z\end{array}\right)$ 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 $\mathrm{M}$. Then,
among the statements:
(I) Trace $(\mathrm{R})=0$
(II) If trace $(\operatorname{adj}(\operatorname{adj}(R))=0$, then $R$ has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Show Answer
Answer (2)
Solution
- $\mathrm{x} \sin \theta=\mathrm{y} \sin \left(\theta+\frac{2 \pi}{3}\right)=\mathrm{z} \sin \left(\theta+\frac{4 \pi}{3}\right) \neq 0$
$\Rightarrow \mathrm{x}, \mathrm{y}, \mathrm{z} \neq 0$
Also,
$$ \begin{aligned} & \sin \theta+\sin \left(\theta+\frac{2 \pi}{3}\right)+\sin \left(\theta+\frac{4 \pi}{3}\right)=0 \forall \theta \in R \ & \Rightarrow \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0 \ & \Rightarrow x y+y z+z x=0 \end{aligned} $$
(i) $\operatorname{Trace}(\mathrm{R})=\mathrm{x}+\mathrm{y}+\mathrm{z}$
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) $\operatorname{Adj}(\operatorname{Adj}(\mathrm{R}))=|\mathrm{R}| \mathrm{R}$
Trace $(\operatorname{Adj}(\operatorname{Adj}(\mathrm{R})))$
$=x y z(x+y+z) \neq 0$
Statement (ii) is also False
