Question

If $\sqrt{-1}=i,$ and $\omega$ is a nonreal cube root of unity then the value of
$\left|\begin{array}{ccc}1& {\omega }^2& 1+i+{\omega }^2\\ -i& -1& -1-i+\omega \\ 1-i& {\omega }^2-1& -1\end{array}\right|$ is equal to

• A. $1$
• B. $i$
• C. $\omega$
• D. $0$

Answer 1

The correct option is D $0$

$\left|\begin{array}{ccc}1& {\omega }^2& 1+i+{\omega }^2\\ -i& -1& -1-i+\omega \\ 1-i& {\omega }^2-1& -1\end{array}\right|$
$\left|\begin{array}{ccc}1& {\omega }^2& i-\omega \\ -i& -1& -1-i+\omega \\ 1-i& {\omega }^2-1& -1\end{array}\right|$
$R_1\to R_1+R_3$
$\left|\begin{array}{ccc}-i& -1& -1-i+\omega \\ -i& -1& -1-i+\omega \\ 1-i& {\omega }^2-1& -1\end{array}\right|$
$=0$