`2x^(2) + 3x - alpha - 0 " has roots "-2 and beta " while the equation "x^(2) - 3mx + 2m^(2) = 0 " has both roots positive, where " alpha gt 0 and beta gt 0.`
If `alpha and beta` are the roots of the equation `1+x+x^(2) = 0`, then the matrix product `[{:(1,beta),(alpha,alpha):}],[{:(alpha,beta),(1,beta):}]` is equal to

Options

A. `[{:(1,1),(1,2):}]`
B. `[{:(-1,-1),(-1,2)]`
C. `[{: (1,-1),(-1,2):}]`
D. `[{: (-1,-1),(-1,-2):}]`

Answer

B

Explanation

`alpha, beta " are the roots of the equation" 1+x+x^(2) =0.`
`1+x+x^(2) = 0 rArr x^(2) + x+1=0`
Solving for x,`x= (-b pm sqrt(b^(2)-4ac))/(2a) = (-1 pm sqrt(1^(2)-4(1)(1)))/(2(1))`
` = (-1 pm sqrt(-3))/2 = (-1 pm sqrt(3i))/2`
i.e., `alpha = omega, beta = omega^(2)`
`[{:(1" " beta),(alpha" "alpha):}][{:(alpha" "beta) ,(1" "beta):}]=[{:(alpha+beta" "beta+beta^(2)),(alpha^(2)+ alpha" "alpha beta + alphabeta):}]`
=`{:[(omega+omega^(2)" "omega^(2) +omega^(4)),(omega^(2)+omega" "omega^(3)+omega^(3))]:}`
`[{:(-1" "omega^(2)+omega),(-1" "2omega^(3))]:}`
`[{:(-1,-1),(-1,2):}]`