Question

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

• A. $-2\omega$
• B. $-3\omega^2$
• C. $-1$
• D. $0$

Answer 1

Answer: (B)

$\begin{vmatrix}1+\omega&\omega^{2}&-\omega\\ 1+\omega^{2}&\omega&-\omega^{2}\\ \omega+\omega^{2}&\omega& \omega^{2}\end{vmatrix}$
$=\begin{vmatrix}0&\omega^{2}&-\omega\\ 0&\omega&-\omega^{2}\\ -1+\omega&\omega&-\omega^{2}\end{vmatrix}$
$\left[\because C_{1} \rightarrow C_{1}+C_{2}\right]$
$=(-1+\omega)\left(-\omega^{4}+\omega^{2}\right)=(\omega-1)\left(\omega^{2}-\omega\right)$
$=\omega^{3}-\omega^{2}-\omega^{2}+\omega=-3 \omega^{2}$