Let $\omega = -\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}$. Then the value of the determinant $\begin{vmatrix}1 & 1 & 1\\1 & -1-\omega^2 & \omega^2\\1 & \omega^2 & \omega\end{vmatrix}$ is
Step-by-Step Solution
Key Concept: The key simplification is $-1-\omega^2=\omega$ (from $1+\omega+\omega^2=0$), which turns the determinant into a DFT-like matrix whose expansion collapses via $\omega^4=\omega$.
**Step 1: Simplify the (2,2) entry**
$-1-\omega^2 = -(1+\omega^2) = -(-\omega) = \omega$. So the matrix becomes $\begin{vmatrix}1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega\end{vmatrix}$.
**Step 2: Expand along row 1**
$\det = 1\cdot(\omega^2-\omega^4) - 1\cdot(\omega-\omega^2) + 1\cdot(\omega^2-\omega) = (\omega^2-\omega)-(\omega-\omega^2)+(\omega^2-\omega) = 3(\omega^2-\omega)$.
**Step 3: Factor**
$3(\omega^2-\omega) = 3\omega(\omega-1)$.
Correct Answer: 2