Let `omega` be a complex cube root of unity with `omega!=1a n dP=[p_(i j)]` be a `nxxn` matrix withe `p_(i j)=omega^(i+j)dot` Then `p^2!=O` , when `n=`
a.`57`
b. `55`
c. `58`
d. `56`

If omega is a complex nth root of unity, then underset(r=1)overset(n)(ar+b)omega^(r-1) is equal to

If omega is the complex cube root of unity then |[1,1+i+omega^2,omega^2],[1-i,-1,omega^2-1],[-i,-i+omega-1,-1]|=

If omega is an imaginary cube root of unity, then (1+omega-omega^2)^7 is equal to (a) 128omega (b) -128omega (c) 128omega^2 (d) -128omega^2

Let omega be a complex cube root unity with omega!=1. A fair die is thrown three times. If r_1, r_2a n dr_3 are the numbers obtained on the die, then the probability that omega^(r1)+omega^(r2)+omega^(r3)=0 is 1//18 b. 1//9 c. 2//9 d. 1//36

Let G= {1, w, w^(2)} where w is a complex cube root of unity. Then find the inverse of w^(2) . Under usual multiplication.

If z=omega,omega^2w h e r eomega is a non-real complex cube root of unity, are two vertices of an equilateral triangle in the Argand plane, then the third vertex z may be represented by (a) z=1 b. z=0 c. z=-2 d. z=-1

If omega(!= 1) be an imaginary cube root of unity and (1+omega^2)=(1+omega^4), then the least positive value of n is (a) 2 (b) 3 (c) 5 (d) 6

If omega is a complex nth root of unity, then sum_(r=1)^n(ar+b)omega^(r-1) is equal to A.. (n(n+1)a)/2 B. (n b)/(1+n) C. (n a)/(omega-1) D. none of these

Let omega!=1 be cube root of unity and S be the set of all non-singular matrices of the form [(1,a, b), (omega, 1, c), (omega^2, omega, 1)] , where each of a ,b and c is either omegaoromega^2 Then the number of distinct matrices in the set S is a. 2 b. 6 c. 4 d. 8

Let omega != 1 be cube root of unity and S be the set of all non-singular matrices of the form [(1,a,b),(omega,1,c),(omega^2,theta,1)], where each of a,b, and c is either omega or omega^2. Then the number of distinct matrices in the set S is (a) 2 (b) 6 (c) 4 (d) 8