[[1,w,w^(2)],[w,w^(2),1],[w^(2),1,w]]+[[w,w^(2),1],[w^(2),1,w],[w,w^(2),1]][[1],[w],[w^(2)]]=[[0],[0],[0]]

det[[x+w^(2),w,1w,w^(2),1+x1,x+w,w^(2)]]=0

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]]

If omega is a complex cube root of unity, show that [[1 , omega, omega^2], [ omega, omega^2, 1],[ omega^2, 1, omega]] [[1],[ omega],[ omega^2]]=[[0],[ 0],[ 0]]

Given that [(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega)][(k,1,1),(1,1,1),(1,1,1)]=[(0,0,0),(0,0,0),(0,0,0)] then k=

{[(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))]

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):}] where omega is the cube root of unit.

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

(1-w^2+w^4)(1+w^2-w^4)

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]