Let $\omega$ be a complex cube root of unity. If the matrix $A=\left(\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right)$, then the value of $\det (A)$ is:

Solution

1. Observe the sum of the columns of matrix $A$.
2. The sum of each row is $1+\omega +{\omega }^2=0$.
3. Since the sum of columns is zero, the columns are linearly dependent.
4. Hence the answer is (A).