If a r = cos 2 r π 9 + i sin 2 r π 9 , r = 1 , 2 , 3 , … , i = - 1 , then the determinant a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 is equal to :

Step-by-step solution

Given that a r = cos 2 r π 9 + i sin 2 r π 9 , r = 1 , 2 , 3 , … , i = - 1 ⇒ a r   =   e i 2 r π 9 - - - - ( I ) From Euler form e i θ   =   cos θ   +   i   sin θ a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 = e i 2 π 9 e i 4 π 9 e i 6 π 9 e i 8 π 9 e i i 10 π 9 e i 12 π 9 e i 14 π 9 e i i 16 π 9 e i 18 π 9 Taking e i 2 π 9 ,   e i 8 π 9 ,   e i 14 π 9 common form each row = e i