Question

Show with out expanding $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|=0$. Where $\omega$ being cube root of unity.

Answer 1

Now,

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

[$C_1^{\prime }=C_1+C_2+C_3$]

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

$=\left|\begin{array}{ccc}0& \omega & {\omega }^2\\ 0& {\omega }^2& 1\\ 0& 1& \omega \end{array}\right|$ [ Since $\omega$ being cube root of unity then $1+\omega +{\omega }^2=0$]

$=0$.