Question

If $(\omega\,\neq\,1)$ is a cube root of unity , then $\begin{vmatrix} 1 &1+i+\omega^2 &\omega^2 \\\[0.3em] 1-i&-1 & \omega^2-1 \\\[0.3em] -i & -1+\omega-i& -1 \end{vmatrix}=$

• A. 0
• B. 1
• C. i
• D. $\omega$

Answer 1

Answer: (A)

0

Answer 2

$\begin{vmatrix} 1 &1+i+\omega^2 &\omega^2 \\\[0.3em] 1-i&-1 & \omega^2-1 \\\[0.3em] -i & -1+\omega-i& -1 \end{vmatrix}$ Operate $R_1 + R_3 - R_2$ $\begin{vmatrix} 0&0 &0 \\\[0.3em] 1-i&-1 & \omega^2-1 \\\[0.3em] -i & -1+\omega-i& -1 \end{vmatrix}= 0$ [ $\therefore$ $\omega^2 + \omega + 1 = 0]$