if w is a complex cube root to unity then value of
` Delta =|{:(a_(1)+b_(1)w,,a_(1)w^(2)+b_(1),,c_(1)+b_(1)bar(w)),(a_(2)+b_(2)w,,a_(2)w^(2)+b_(2),,c_(2)+b_(2)bar(w)),(a_(3)+b_(3)w,,a_(3)w^(2)+b_(3),,c_(3)+b_(3)bar(w)):}|` is

Options

A. 0
B. -1
C. 2
D. none of these

Answer

A

Explanation

`Delta =|{:(a_(1)+b_(1)w,,a_(1)w^(2)+b_(1),,c_(1)+b_(1)hat(w)),(a_(2)+b_(2)w,,a_(2)w^(2)+b_(2),,c_(2)+b_(2)hat(w)),(a_(3)+b_(3)w,,a_(3)w^(2)+b_(3),,c_(3)+b_(3)hat(w)):}|`
`" Operating " C_(2) to wC_(2) ` we have
`Delta =(1)/(w) |{:(a_(1)+b_(1)w,,a_(1)w^(2)+b_(1),,c_(1)+b_(1)hat(w)),(a_(2)+b_(2)w,,a_(2)w^(2)+b_(2),,c_(2)+b_(2)hat(w)),(a_(3)+b_(3)w,,a_(3)w^(2)+b_(3),,c_(3)+b_(3)hat(w)):}|`
`=(1)/(w)|{:(a_(1)+b_(1)w,,a_(1)+b_(1)w,,c_(1)+b_(1)hat(w)),(a_(2)+b_(2)w,,a_(2)+b_(2)w,,c_(2)+b_(2)hat(w)),(a_(3)+b_(3)w,,a_(3)+b_(3)w,,c_(3)+b_(3)hat(w)):}|`
`=0`