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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
`because`
`A=1/sqrt3[(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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