Solution:

Given

It is also given that \[\omega \] is a complex cube root of unity,

Consider the LHS,

We know that \[1\text{ }+\text{ }\omega \text{ }+\text{ }{{\omega }^{2}}~=\text{ }0\text{ }and\text{ }{{\omega }^{3}}~=\text{ }1\]

Now by simplifying we get,

Again by substituting \[1\text{ }+\text{ }\omega \text{ }+\text{ }{{\omega }^{2}}~=\text{ }0\text{ }and\text{ }{{\omega }^{3}}~=\text{ }1\] in above matrix we get,

Therefore \[LHS\text{ }=\text{ }RHS\]

Hence the proof.