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Let alpha, beta be the roots of equat...

Let ` alpha, beta ` be the roots of equation ` x ^ 2 - x + 1 = 0 ` and the matrix ` A = (1 ) /(sqrt3 ) |{:(1,,1,,1),(1,,alpha,,alpha ^2),(1,,beta,,-beta^ 2):}| ` , the value of det ` (A. A^T)` is

A

` ( 1 ) /(3) `

B

` 1 `

C

` - 1 `

D

` 3 `

Text Solution

Verified by Experts

The correct Answer is:
C
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A = (1)/(sqrt(3)) [{:(1,1,1),(1,-omega,- omega^(2)),(1,-omega^(2),-omega^(4)):}] , det (A.A^(T)) = |A|^(2)`
` = (1)/(3) |{:(1,-1,-1),(1,omega,omega^(2)),(1,omega^(2),omega):}| , because omega^(3) = 1 , 1 + omega + omega^(2) = 0`
` = (1)/(3) (omega^(2) - omega^(4) + omega- omega^(2) + omega - omega^(2))^(2)`
` = (1)/(3) (omega - omega^(2))^(2) " " = (1)/(3) (omega^(2) + omega^(4) - 2omega^(3))`
` = (1)/(3) (-1-2) = - 1 det (A.A^(T)) = - 1`
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