Question

$\omega$ is a complex cube root of unity then $\left|\begin{array}{ccc}1& 1+w& 1+w^2\\ 1+w& 1+w^2& 1\\ 1+w^2& 1& 1+w\end{array}\right|$

• A. $4$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

The correct option is A $4$

We have, $\left|\begin{array}{ccc}1& 1+\omega & 1+{\omega }^2\\ 1+\omega & 1+{\omega }^2& 1\\ 1+{\omega }^2& 1& 1+\omega \end{array}\right|$

Performing $C_1\to C_1+C_2+C_3$

$\Rightarrow$$\left|\begin{array}{ccc}(1+\omega +{\omega }^2)+2& 1+\omega & 1+{\omega }^2\\ (1+\omega +{\omega }^2)+2& 1+{\omega }^2& 1\\ (1+\omega +{\omega }^2)+2& 1& 1+\omega \end{array}\right|$

We know that $1+\omega +{\omega }^2=0$

Taking 2 common from $C_1$

$\Rightarrow$$2\left|\begin{array}{ccc}1& 1+\omega & 1+{\omega }^2\\ 1& 1+{\omega }^2& 1\\ 1& 1& 1+\omega \end{array}\right|$

$\Rightarrow$$R_1\to R_1-R_2$ $R_2\to R_2-R_3$

$\Rightarrow$$2\left|\begin{array}{ccc}0& \omega -{\omega }^2& {\omega }^2\\ 0& {\omega }^2& -\omega \\ 1& 1& 1+\omega \end{array}\right|$

$=2\left[1\right(\omega -{\omega }^2\left)\right(-\omega )-{\omega }^4]$ ............. $({\omega }^3=1)$

$=2[{\omega }^3-{\omega }^2-\omega ]$

$=2[1+1-(1+\omega +{\omega }^2\left)\right]$

$=2[1+1]=4$