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If omega is a complex cube root of uni...

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)]=[(0),( 0 ),(0)]`

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Given ,
`([(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)]`
`(LHS)=`
`([(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+omega^2,omega^2+1),(omega+omega^2,omega^2+1,1+omega),(omega^2+omega,1+omega^2,omega+1)][(1),(omega),(omega^2)]`
`=[(-omega^2,-1,-omega),(-1,-omega,-omega^2),(-1,-omega,omega^2)][(1),(omega),(omega^2)]`
`=[(- omega^2-omega-omega^3),(-1-omega^2-omega^4),(-1-omega-omega^4)]`
`=[(-omega(1+omega+omega^2)),(-1-omega^2-omega^3omega),(-1-omega^2-omega^3omega)]`
`=[(0),(-1-omega^2-omega),(-1-omega^2-omega)]`
`=[(0),( 0 ),(0)]`
Hence `(LHS=RHS)`
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