If w is a complex cube root of unity, then value of a1+b1w a1w2+b1 c1+b1w¯a2+b2w a2w2+b2 c2+b2w¯a3+b3w a3w2+b3 c3+b3w¯ is
0
-1
2
none of these
let ∆= a1+b1w a1w2+b1 c1+b1w¯a2+b2w a2w2+b2 c2+b2w¯a3+b3w a3w2+b3 c3+b3w¯
Operating C2→wC2, we have
Δ=1wa1+b1wa1w3+b1wc1+b1w¯a2+b2wa2w3+b2wc2+b2w¯a3+b3wa3w3+b3wc3+b3w¯=1wa1+b1wa1+b1wc1+b1w¯a2+b2wa2+b2wc2+b2w¯a3+b3wa3+b3wc3+b3w¯ ∵ω3=1=0