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 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 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)]

(2+omega+omega^2)^3+(1+omega-omega^2)^8-(1-3omega+omega^2)^4=1

Evaluate |(1,omega,omega^2),(omega,omega^2,1),(omega^2,omega, omega)| where omega is cube root of unity.

If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |[x/(1+omega), y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[(z)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

Without expanding at any stage, prove that |{:(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega):}| = 0

if omega!=1 is cube root of unity and x+y+z != 0 then |[x/(1+omega),y/(omega+omega^2),z/(omega^2+1)],[y/(omega+omega^2),z/(omega^2+1),x/(1+omega)],[z/(omega^2+1),x/(1+omega),y/(omega+omega^2)]| =0 if

Prove the following (1- omega + omega^(2)) (1 + omega- omega^(2)) (1 - omega- omega^(2))= 8

Prove that (1-omega+omega^(2))(1-omega^(2)+omega^(4))(1-omega^(4)+omega^(8))(1-omega^(8)+omega^(16)).... to 2^(n) factors = 2^(2n) .

If 1.omega, omega^2 are the roots of unity, then Delta=|(1,omega^n,omega^(2n)),(omega^n,omega^(2n),1),(omega^(2n),1,omega^n)| is equal to