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.

Evaluate the following determinates |{:(1,omega),(omega,-omega):}| where omega is the cube root of unity

Using the property of determinants and without expanding prove the following |{:(1,omega^n,omega^(2n)),(omega^(2n),1,omega^n),(omega^n,omega^(2n),1):}|=0 where omega is a cube root of unity

If 1,omega,omega^(2),...omega^(n-1) are n, nth roots of unity, find the value of (9-omega)(9-omega^(2))...(9-omega^(n-1)).

If omega(ne1) is a cube root of unity, then (1-omega+omega^(2))(1-omega^(2)+omega^(4))(1-omega^(4)+omega^(8)) …upto 2n is factors, is

If omega is a non-real complex cube root of unity, find the values of the following. (i) omega^(1999) (ii) omega^(998) (iii) ((-1+isqrt(3))/2)^(3n+2),ninNand i=sqrt(-1) (iv) (1+omega)(1+omega )^(2)(1+omega)^(4)(1+omega)^(8) ...upto 2n factors (v) ((alpha+betaomega+gammaomega^(2)+deltaomega^(2))/(beta+alphaomega^(2)+gammaomega+deltaomega))," where "alpha,beta,gamma,delta, in R (vi) 1*(2-omega)(2-omega^(2))+2*(3-omega)(3-omega^(2))+3*(4-omega)(4-omega^(2))+...+...+(n-1)*(n-omega)(n-omega^(2))

Let omega be a complex number such that 2omega+1=z where z=sqrt(-3) . If |{:(1,1,1),(1,-omega^(2)-1,omega^(2)),(1,omega^(2),omega^(7))|=3k , then k is equal to

Given 3[{:(x,y),(z,w):}]=[{:(x,6),(-1,2w):}]+[{:(4,x+y),(z+w,3):}] , find the values of x,y,z and w.

Let omega ne 1 be a complex cube root of unity. If (3-3omega+2omega^(2))^(4n+3) + (2+3omega-3omega^(2))^(4n+3)+(-3+2omega+3omega^(2))^(4n+3)=0 , then the set of possible value(s) of n is are

Which of the following can not be valid assignment of probabilities for outcomes of sample Space S={omega_(1),omega_(2),omega_(3),omega_(4),omega_(5),omega_(6),omega_(7)}