Q. $\left| \begin{matrix} \alpha & - \beta & 0 \\ 0 & \alpha & \beta \\ \beta & 0 & \alpha \end{matrix} \right|$ =0, then
A) $\frac{\alpha}{\beta}$ is one of the cube roots of unity
B) $\alpha\$is one of the cube roots of unity
C) $\beta$ is one of the cube roots of unity
D) $\alpha\beta$ is one of the cube roots of unity
Explanation
(a)
$\left| \begin{matrix}
\alpha & - \beta & 0 \\
0 & \alpha & \beta \\
\beta & q & x
\end{matrix} \right| = 0 \Rightarrow \alpha^{3} - \beta^{3} = 0$
$\Rightarrow \left( \frac{\alpha}{\beta} \right)^{3} = 1 \Rightarrow \frac{\alpha}{\beta}$ is one of the cube roots of unity.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.