Question

If $\omega$ is the cube root of unity, then $\left| \begin{matrix} 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega \end{matrix} \right|$=

• A. 1
• B. 0
• C. $\omega$
• D. $\omega^{2}$

Answer 1

Answer: (B)

0

Answer 2

$\left| \begin{matrix} 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega \end{matrix} \right| = \left| \begin{matrix} 1 + \omega + \omega^{2} & \omega & \omega^{2} \\ 1 + \omega + \omega^{2} & \omega^{2} & 1 \\ 1 + \omega + \omega^{2} & 1 & \omega \end{matrix} \right|$

$= \left| \begin{matrix} 0 & \omega & \omega^{2} \\ 0 & \omega^{2} & 1 \\ 0 & 1 & \omega \end{matrix} \right| = 0$.