Question

The value of the determinant $\left|\begin{array}{ccc}1& {\omega }^3& {\omega }^5\\ {\omega }^3& 1& {\omega }^4\\ {\omega }^5& {\omega }^4& 1\end{array}\right|$ , where $\omega$ is an imaginary cube root of unity,is

• A. $(1-\omega {)}^2$
• B. 3
• C. -3
• D. none of these

Answer 1

The correct option is D none of these

$\left|\begin{array}{ccc}1& {\omega }^3& {\omega }^5\\ {\omega }^3& 1& {\omega }^4\\ {\omega }^5& {\omega }^4& 1\end{array}\right|=\left|\begin{array}{ccc}1& 1& {\omega }^2\\ 1& 1& \omega \\ {\omega }^2& \omega & 1\end{array}\right|\Rightarrow {\omega }^2-2\omega +1$ $(\because {\omega }^3=1and1+\omega +{\omega }^2=0)$
$\Rightarrow (\omega -1{)}^2$