Simplify `{[[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]]`

|[1,omega,omega^2] , [omega, omega^2,1] , [omega^2,1,omega]|=0

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)]

det [[1, omega, omega^(2) omega, omega^(2), 1omega^(2), 1, omega]]

det [[1, omega, omega^(2) omega, omega^(2), 1omega^(2), 1, omega]] =

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

If omega is a complex cube root of unity,show 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)]=[000]

omega(1+omega-omega^(2))=-2

(1-omega+omega^(2))(1+omega-omega^(2))=4