Question

Evaluate the determinant 1 w w2 , w w2 1, w2 1 w. where w is a cube root of unity.

Answer 1

Answer:

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

Step-by-step explanation:

Given:

$\omega$ is a cube root of unity

Then,

$\bf\:{\omega}^3=1$ and

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

Now

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

$=\left|\begin{array}{ccc}1+\omega+{\omega}^2&\omega&{\omega}^2\\1+\omega+{\omega}^2&{\omega}^2&1\\1+\omega+{\omega}^2&1&\omega\end{array}\right|$ $C_1\implies\,C_1+C_2+C_3$

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

$=0$

Answer 2

Answer:

Heya mate,

Refer to the attachment for your answer