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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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ARIHANT MATHS-MATRICES -Exercise (Questions Asked In Previous 13 Years Exam)
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  5. If A^(2)-A+I=O, then A^(-1) is equal to

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  7. Let A = [(1,0,0), (2,1,0), (3,2,1)], and U1, U2 and U3 are columns of ...

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  12. Let A and B be 3xx3 matrtices of real numbers, where A is symmetric, "...

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  14. Let A be a 2xx2 matrix with real entries. Let I be the 2xx2 identi...

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  15. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

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  16. Let A be the set of all 3xx3 symmetric matrices all of whose either 0 ...

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  17. The number of 3xx3 matrices A whose are ether 0 or 1 and for which t...

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  18. Let A be a 2xx2 matrix Statement -1 adj (adjA)=A Statement-2 abs(a...

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  19. The number of 3xx3 matrices a whose entries are either 0 or 1 and for ...

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  20. Let P be an odd prime number and T(p) be the following set of 2xx2 mat...

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  21. Let p be an odd prime number and T(P) be the following set of 2xx2 m...

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