Question

If a is a complex cube root 2, find
$\Delta =\left|\begin{array}{ccc}1& 2a& 1\\ a^2& 1& 4a^2\\ 3& 2a& 1\end{array}\right|$

Answer 1

$△=\left|\begin{array}{ccc}1& 2a& 1\\ a^2& 1& 4a^2\\ 3& 2a& 1\end{array}\right|$

$△=1(1-8a^3)-2a(a^2-12a^2)+1(2a^3-3)$

$△=1-8a^3-2a^3+24a^3+2a^3-3$

$△=-2+16a^3$

a is cube root of infinity

$\Rightarrow a=\sqrt[3]{32}$

$△-1+16{\left(\sqrt[3]{32}\right)}^3$

$△=-2+32$

$△=30$

Correct answer is $30$