Question

If $w$ be the complex cube root of unity then value of
$\left|\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right|$ is

$\left(a\right)0\left(b\right)1$
$\left(c\right)w\left(d\right)w^2$

Answer 1

Lets take given determinant as $\Delta$
$\Delta =$ $\left|\begin{array}{ccc}1& w& w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right|$

Perform $R_1\to R_1+R_2+R_3$
$\Delta =$ $\left|\begin{array}{ccc}1+w+w^2& 1+w+w^2& 1+w+w^2\\ w& w^2& 1\\ w^2& 1& w\end{array}\right|$

We know $1+w+w^2=0$
$\Delta =$ $\left|\begin{array}{ccc}0& 0& 0\\ w& w^2& 1\\ w^2& 1& w\end{array}\right|$

As all $R_1$ elements are zeros
$\therefore \Delta =0$
Answer: $\left(a\right)$