Question

If $\omega$ is a cube root of unity, then the value of determinant $\begin{vmatrix}1+\omega&\omega^{2}&\omega \\\ \omega^{2} + \omega &-\omega &\omega^{2} \\\ 1+\omega^{2} &\omega &\omega^{2} \end{vmatrix}$

• A. $1 + \omega$
• B. $\omega - 1$
• C. 0
• D. $\omega^2$

Answer 1

Answer: (B)

$\omega - 1$

Answer 2

Applying $C_1 \to C_1 +C_2$
$\begin{vmatrix}1+\omega+\omega^{2} &\omega^{2}&\omega \\\ \omega^{2} &-\omega &\omega^{2} \\\ 1+\omega + \omega^{2} &\omega &\omega^{2} \end{vmatrix} = \begin{vmatrix}0&\omega^{2}&\omega \\\ \omega^{2} &-\omega &\omega^{2} \\\ 0&\omega &\omega^{2} \end{vmatrix}$
$\ \ \ \ \ \ \ \ \ \ [ \because \ 1+ \omega + \omega^2 = 0]$
Expanding along $C_1$? we get $- \omega^2 (\omega^4 -\omega^2)$
$= - \omega^6 +\omega^4 = \omega - 1\ \ \ \ \ \ \ \ \ \ \ [\because \ \ \omega^3 = 1]$