Question

If `omega`is an imaginary cube root of unity, then the value of the determinant `|(1+omega,omega^2,-omega),(1+omega^2,omega,-omega^2),(omega+omega^2,omega,-omega^2)|`

Answer 1

Correct Answer - D
Using `1 + omega + omega^(2) = 0`, we get
`|(1 + omega,omega^(2),-omega),(1 + omega^(2),omega,- omega^(2)),(omega^(2) + omega,omega,- omega^(2))|`
`= |(1 + omega + omega^(2),omega^(2),- omega),(1 + omega^(2) + omega,omega,- omega^(2)),(omega^(2) + 2 omega,omega,- omega^(2))| " " ["Applying" C_(1) rarr C_(1) + C_(2)]`
`= |(0,omega^(2),-omega),(0,omega,-omega^(2)),(omega -1,omega,-omega^(2))|`
`= (omega -1)|(omega^(2),-omega),(omega,- omega^(2))|`
`= (omega -1) (-omega^(4) + omega^(2)) = (omega -1)(- omega + omega^(2))`
`= - omega^(2) + omega^(3) + omega - omega^(2) = -omega^(2) + (1 + omega) - omega^(2) = -3 omega^(2)`