Let `omega = - (1)/(2) + i (sqrt3)/(2)`, then the value of the determinant `|(1,1,1),(1,-1- omega^(2),omega^(2)),(1,omega^(2),omega^(4))|`, is

Options

A. `3omega`
B. `3omega(omega-1)`
C. `3omega^(2)`
D. `3omega(1-omega)`

Answer

B

Explanation

We have,
`Delta=|{:(1,1,1),(1,-1-omega^(2),omega^(2)),(1,omega^(2),omega^(4))|`
`rArr Delta=|{:(1,0,0),(1,-2-omega^(2),omega^(2)-1),(1,omega^(2)-1,omega-1):}|` Applying `C_(2) rArr C_(2)-C_(1)`,
`rArr Delta=-(2omega+omega^(3)-2-omega^(2))-(omega^(4)-2omega^(2)+1)`
`rArr Delta=-(2omega-1-omega^(2))-(omega-2omega^(2)+1)`
`rArr Delta=-(2omega+omega)-(-3omega^(2))=-3omega+3omega^(2)=3omega(omega-1)`