Solveeit Logo
Login

Question

Question: If \(\omega\)is a complex cube root of unity, then the determinant\(\left| \begin{matrix} 2 & 2\omeg...

If ω\omegais a complex cube root of unity, then the determinant∣22ω−ω21111−10∣=\left| \begin{matrix} 2 & 2\omega & - \omega^{2} \\ 1 & 1 & 1 \\ 1 & - 1 & 0 \end{matrix} \right| =.

A

0

B

1

C

– 1

D

None of these

Answer

0

Explanation

Solution

Δ≡∣22ω−ω21111−10∣=∣2+2ω+2ω22ω−ω21+1−2111−1−0−10∣\Delta \equiv \left| \begin{matrix} 2 & 2\omega & - \omega^{2} \\ 1 & 1 & 1 \\ 1 & - 1 & 0 \end{matrix} \right| = \left| \begin{matrix} 2 + 2\omega + 2\omega^{2} & 2\omega & - \omega^{2} \\ 1 + 1 - 2 & 1 & 1 \\ 1 - 1 - 0 & - 1 & 0 \end{matrix} \right|

(C1→C1+C2−2C3)(C_{1} \rightarrow C_{1} + C_{2} - 2C_{3})

= ∣02ω−ω20110−10∣=0\left| \begin{matrix} 0 & 2\omega & - \omega^{2} \\ 0 & 1 & 1 \\ 0 & - 1 & 0 \end{matrix} \right| = 0.