Question

Evaluate the following determinants without expansion as far as possible.
$\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|$
Where $1,\omega ,{\omega }^2$ are the cube roots of unity.

Answer 1

Consider the given determinant.

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

Here,

$1+\omega +{\omega }^2=0$

Apply $C_1\to C_1+C_2+C_3$.

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

$\Delta =\left|\begin{array}{ccc}0& \omega & {\omega }^2\\ 0& {\omega }^2& 1\\ 0& 1& \omega \end{array}\right|$

Therefore,

$\Delta =0$