Home
Class 12
MATHS
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 equatin `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 - a ^(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, 2 omega ^(2), -1),(-2 omega, -3, 0):}]," "|AB|=-3`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let alpha be a root of the equation x ^(2) + x + 1 = 0 and the matrix A = ( 1 ) /(sqrt3) [{:( 1,,1,,1),( 1,, alpha ,, alpha ^(2)), ( 1 ,, alpha ^(2),, alpha ^(4)):}] then the matrix A ^( 31 ) is equal to :

If alpha is a roots of equation x^(2)+x+1=0 and A=(1)/(sqrt3)[{:(,1,1,1),(,1,alpha,alpha^(2)),(,1,alpha^(2),alpha):}] then A^(31) equal to:

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

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 :

If alpha and beta be the roots of the equation x^(2)+3x+1=0 then the value of ((alpha)/(1+beta))^(2)+((beta)/(alpha+1))^(2)

Let alpha and beta be the roots of the equation x^(2) + x + 1 = 0 . Then, for y ne 0 in R. |{:(y+1, alpha,beta), (alpha, y+beta, 1),(beta, 1, y+alpha):}| is

If alpha and beta be the roots of the equation x^2-1=0 , then show that. alpha+beta=(1)/(alpha)+(1)/(beta)

If alpha is a root of the equation 4x^(2)+2x-1=0 then the other root is given by O -2 alpha-1 O 4 alpha^(2)+alpha-1 O 4 alpha^(3)-3 alpha O 4 alpha^(2)-3 alpha

If alpha, beta are the roots of the equation x^(2)-ax+b=0 .then the equation whose roots are 2 alpha+(1)/(beta), 2 beta+(1)/(alpha) is