Elements of a matrix A of order `10xx10` are defined as `a_("ij")=omega^(i+j)` (where `omega` is imaginary cube root of unity), then trace (A) of the matrix is

Options

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

Answer

D

Explanation

tr `(A) = sum_(i=j=1)^(10) a_(ij) = sum _(i=j=1) ^(10) omega^(i+j) = sum_(i=1) ^(10) omega ^(2i) `
`= omega^(2)+omega^(4) + omega^(6) + omega^(8) + ... + omega^(20) `
`= (omega^(2)+omega + 1 )+( omega^(2) + omega+1)+(omega^(2)+omega+1) + omega^(20) `
`= 0 + 0 + 0+ omega ^(2) = omega^(2)`