If `omega` is a non-real cube root of unity and `n` is not a multiple of 3, then `=|1omega^nomega^(2n)omega^(2n)1omega^nomega^nomega^(2n)1|` is equal to (a) 0 (b) `omega` (c) `omega^2` (d) 1

Options

A. 0
B. `omega`
C. `omega^(2)`
D. 1

Answer

A

Explanation

Applying `C_(1) rarr C_(1) + C_(2) + C_(3)`, we get
`Delta = (1 + omega^(n) + omega^(2n)) |(1,omega^(n),omega^(2n)),(1,1,omega^(n)),(1,omega^(2n),1)|`
`rArr Delta = 0 xx |(1,omega^(n),omega^(2n)),(1,1,omega^(n)),(1,omega^(2n),1)| " " [( :' 1 + omega^(n) + omega^(2n) = 0),("when " n != 3k"," k in N)]`
`rArr Delta = 0`