What is the value of `|{:(1,omega,2omega^(2)),(2,2omega^(2),4omega^(3)),(3,3omega^(3),6omega^(4)):}|`, where `omega` is the cube root of unity ?

Options

A. 0
B. 1
C. 2
D. 3

Answer

A

Explanation

Consider `|{:(1,omega,2omega^(2)),(1,omega^(2),4omega^(3)),(3,3omega^(3),6omega^(4)):}|=|{:(1,omega,2omega^(2)),(2,2omega^(2),4),(3,3,6omega):}|" "(because omega^(3)=1 and omega^(4)=omega)`
`=2xx3|{:(1,omega,2omega^(2)),(1,omega^(2),2),(1,1,2omega):}|`
`=6[1(2omega^(3)-2)-omega(2omega-2)+2omega^(2)(1-omega^(2))]`
`=6[0-2omega^(2)-2omega+2omega^(2)-2omega]=0`