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Elements of a matrix A of order 10xx10 a...

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

A

0

B

1

C

3

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
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)`
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