|[1,w,w^(2)],[w,w^(2),1],[w^(2),1,w]|=

If w is a complex cube root of unity, then the value of the determinant Delta = [(1,w,w^(2)),(w,w^(2),1),(w^(2),1,w)] , is

Without expanding at any stage, prove that the value of each of the following determinants is zero. (1) |[0,p-q,p-r],[q-p,0,q-r],[r-p,r-q,0]| (2)|[41,1,5],[79,7,9],[29,5,3]| (3)|[1,w,w^2],[w,w^2,1],[w^2,1,w]| , where w is cube root of unity

Without expanding at any stage, prove that the value of each of the following determinants is zero. (1) |[0,p-q,p-r],[q-p,0,q-r],[r-p,r-q,0]| (2) |[41,1,5],[79,7,9],[29,5,3]| (3) |[1,w,w^2],[w,w^2,1],[w^2,1,w]| , where w is cube root of unity

Without expanding at any stage, prove that the value of each of the following determinants is zero. (1) |[0,p-q,p-r],[q-p,0,q-r],[r-p,r-q,0]| (2) |[41,1,5],[79,7,9],[29,5,3]| (3) |[1,w,w^2],[w,w^2,1],[w^2,1,w]| , where w is cube root of unity

Without expanding at any stage, prove that the value of each of the following determinants is zero. (1) |[0,p-q,p-r],[q-p,0,q-r],[r-p,r-q,0]| (2) |[41,1,5],[79,7,9],[29,5,3]| (3) |[1,w,w^2],[w,w^2,1],[w^2,1,w]| , where w is cube root of unity

Select and write the correct answer from the given alternatives in each of the following:If A=[[1,w,w^2],[w,w^2,1],[w^2,1,w]] and B=[[w,w^2,1],[w^2,1,w],[w,w^2,1]] , and C=[[1],[w],[w^2]] where w is the complex cube root of 1,then (A+B)C is equal to

If w is a complex cube root of unity, show that ([[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]]

Prove that |[1,w^2,w^2],[w^2,1,w],[w^2,w,1]|=-3w where w is a cube root of unity.

Let omega be the complex number cos (2 pi)/(3)+i sin (2 pi)/(3) . Then the number of distinct complex number z satisfying [[z+1,omega,omega^(2)],[omega,z+omega^2,1],[omega^(2),1,z+omega]] = 0 is equal to