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Let alpha be a root of the equation x^2-...

Let `alpha` be a root of the equation `x^2-x+1=0` , and the matrix `A=[(1,1,1),(1,alpha,alpha^2),(1,alpha^2,alpha^4)]` and matrix `B=[(1,-1,-1),(1,alpha,-alpha^2),(-1,-alpha^2,-alpha^4)]` then the value of |AB| is :

A

1

B

`-1`

C

`3`

D

`-3`

Text Solution

Verified by Experts

The correct Answer is:
D
The roots of the equation `x^2-x+1=0` are `-omega,-omega^2`
`alpha=-omega`
`AB=[(1,1,1),(1,alpha,alpha^2),(1,alpha^2,alpha^4)] [ (1,-1,-1),(1,alpha,-alpha^2),(-1,-alpha^2,-alpha^4)]`
`AB=[(1+1-1,-1+alpha-alpha^2,-1-alpha^2-alpha^4),(1+alpha-alpha^2,-1+alpha^2-alpha^4,-1-alpha^3-alpha^6),(1+alpha^2-alpha^4, -1+alpha^3-alpha^6,-1-alpha^4-alpha^8)]`
Substituting `alpha=-omega` and simplifying , we get
`AB=[(1,0,0),(2,2omega^2,-1),(-2omega, -3,0)]` , |AB|=3
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