Elements of a matrix A of order 10 x 10 are defined as `a_(ij)=omega^(i+j)` (where omega is cube root unity), then tr(A) of matrix is

Options

A. 0
B. 1
C. 3
D. none of these

Answer

D

Explanation

`tr (A)= sum_(i=j) a_("ij")`
`=(a_(11)+a_(22)+a_(33)+...+a_(1010))`
`=(omega^(2)+omega^(4)+omega^(6)+...+ omega^(20))`
`=omega^(2) (1+omega^(2)+omega^(4) +...+ omega^(18))`
`=omega^(2) [(1+omega+omega^(2))+...+ (1+omega+omega^(2))+1]`
`=omega^(2)xx1`
`=omega^(2)xx1`
`=omega^(2)`