Complex Numbers 5 Question 13
14.
Let $\omega$ be the complex number $\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}$. Then the number of distinct complex number $z$ satisfying $\left|\begin{array}{ccc}z+1 & \omega & \omega^{2} \\ \omega & z+\omega^{2} & 1 \\ \omega^{2} & 1 & z+\omega\end{array}\right|=0$ is equal to … .
(2010)
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Answer:
Correct Answer: 14. (1)
Solution:
- Let
$ A=\begin{vmatrix} 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega \end{vmatrix} $
Now, $\quad A^{2}=$ $ \begin{vmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{vmatrix} $ and $\operatorname{Tr}(A)=0,|A|=0$
$A^{3}=0$
$ \begin{aligned} & \Rightarrow \quad\left|\begin{array}{ccc} z+1 & \omega & \omega^{2} \\ \omega & z+\omega^{2} & 1 \\ \omega^{2} & 1 & z+\omega \end{array}\right|=[A+z l]=0 \\ & \Rightarrow \quad z^{3}=0 \end{aligned} $
$\Rightarrow z=0$, the number of $z$ satisfying the given equation is 1 .
