Simplify `|[1,omega^6,omega^8],[omega^6,omega^3,omega^7],[omega^8,omega^7,omega]|`

If omega is the cube root of unity then find the value of |[1,omega^(6),omega^(8)],[omega^(6),omega^(3),omega^(7)],[omega^(8),omega^(7),1]|

Solve: /_\ = |[omega^(3),omega^(4),omega^(5)],[omega^(6),omega^(8),omega^(2)],[omega^(7),omega^(9),omega]|

|[omega+omega^(2),1,omega],[omega^(2)+1,omega^(2),1],[1+omega,omega,omega^(2)]|

If omega is cube roots of unity, prove that {[(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega)]+[(omega,omega^2,1),(omega^2,1,omega),(omega,omega^2,1)]} [(1),(omega),(omega^2)]=[(0),(0),(0)]

If omega is cube root of unit,then value of | omega^(1)quad omega^(4)quad omega^(8) determinant det[[omega^(1),omega^(8),omega^(8)omega^(4),omega^(8),1omega^(8),1,omega^(4)]] is

{[(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega)] + [(omega,omega^(2),1),(omega^(2),1,omega),(omega,omega^(2),1)]} [(1),(omega),(omega^(2))]