Let Z 1 , Z 2 , Z 3 be three non zero complex numbers such that a = Z 1 , b = Z 2 , c = Z 3 . if the determinant a b c b c a c a b = 0 , then

Step-by-step solution

Let Z 1 , Z 2 , Z 3 be three non zero complex numbers such that a = Z 1 , b = Z 2 , c = Z 3 . Given, a b c b c a c a b = 0 a b c - a 2 - b b 2 - a c + c a b - c 2 = 0 a b c - a 3 - b 3 + a b c + a b c - c 3 = 0 3 a b c - a 3 - b 3 - c 3 = 0 a 3 + b 3 + c 3 - 3 a b c = 0 We know that: a 3 + b 3 + c 3 - 3 a b c = a + b + c a 2 + b 2 + c 2 - a b - b c - c a a + b + c a 2 + b 2 + c 2 - a b - b c - c a = 0 Multiplying and dividing by 2 1 2 a + b + c 2 a 2 + 2 b 2 + 2 c 2 - 2 a b - 2 b c - 2 c a = 0 1 2 a + b + c a - b 2