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Let alpha be a root of the equation x ^(...

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 vlaue 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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