Let A = 1 a 3 0 0 1 b 3 c 3 0 1 where a , b , c ∈ R . If the sum of all non-real roots of the equation A - x I = 0 is k - m a b c , ∀ k , m ∈ Z , then the value of k + m is equal to

Step-by-step solution

A - n I = 1 - n a 3 0 0 1 - n b c 3 0 1 - n = 0 ⇒ 1 - n 1 - n 2 - a 3 - b 3 c 3 = 0 ⇒ - n - 1 3 + a 3 b 3 c 3 = 0 ⇒ n - 1 3 = a b c 3 ⇒ n = 1 + a b c , 1 + a b c w , 1 + a b c w 2 Sum of non real roots = 2 + a b c w + w 2 = 2 - a b c k = 2 , m = 1 ⇒ k + m = 3