`|(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega)|`

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)]

Which of the following is a non singular matrix? (A) [(1,a,b+c),(1,b,c+a),(1,c,a+b)] (B) [(1,omega, omega^2),(omega, omega^2,1),(omega^2,1,omega)] where omega is non real and omega^2=1 (C) [(1^2,2^2,3^2),(2^2,3^2,4^2),(3^2,4^2,5^2)] (D) [(0,2,-3),(-2,0,5),(3,-5,0)]

{[(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))]

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=

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

Find the values of the following determinants : (i) |{:(12,3,4),(16,5,0),(21,-1,2):}| (ii) |{:(256,240,219),(240,225,198),(219,198,181):}| (iii) |{:(17,19,24),(6,8,13),(-1,1,6):}| (iv) |{:(67,19,21),(39,13,14),(81,24,26):}| (v) |{:(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),omega,1):}|" where "omega "is a cube root of unity". (iv) |{:(1,x,y),(0,(2pi)/5,sin(pi)/10),(0,sin((2pi)/5),cos(pi)/10):}|

Show that det ([1,omega,omega^(2)omega,omega^(2),1omega,omega^(2),1omega^(2),1,omega])=0 where omega is the non-real cube root of unity =0 where omega is