Question

If $n \neq 3k$ and 1, $\omega$, $\omega^{2}$ are the cube roots of unity, then $\Delta = \left| \begin{matrix} 1 & \omega^{n} & \omega^{2n} \\ \omega^{2n} & 1 & \omega^{n} \\ \omega^{n} & \omega^{2n} & 1 \end{matrix} \right|$ has the value

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

Answer 1

Answer: (A)

0

Answer 2

Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3}$, we get

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

$(\because 1 + \omega^{n} + \omega^{2n} = 0\text{if}nis\text{notmultipleof}3)$