If $\alpha $ is a non-real cube root of $-2, $ then the value of

$\begin{vmatrix} 1 & 2\alpha & 1\\ \alpha^2 & 1 & 3\alpha^2 \\2 & 2\alpha & 1 \end{vmatrix} ,$ is

-13

The correct answer is option (1) : -13

We have, $\alpha = (-2)^{1/3} w$or, $\alpha = (-2)^{1/3} w^2$

$∴\alpha^3=-2$

Now, $Δ=\begin{vmatrix} 1 & 2\alpha & 1\\ \alpha^2 & 1 & 3\alpha^2 \\2 & 2\alpha & 1 \end{vmatrix}$

$Δ=\begin{vmatrix} 1 & 2\alpha & 1\\ \alpha^2 & 1 & 3\alpha^2 \\1 & 0 & 0 \end{vmatrix}$     [Applying $R_3→R_3-R_1$]

$Δ=6\alpha^3 -1 $       [On expanding along $R_3$ ]

$Δ=-13 $    $[∵\alpha^3= -2]$