Elements of a matrix A or orddr 10xx10 a...

Elements of a matrix `A` or orddr `10xx10` are defined as `a_(i j)=w^(i+j)` (where `w` is cube root of unity), then trace `(A)` of the matrix is a.`0` b. `1` c. `3` d. none of these

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) 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)`

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