If a , b , c are non - zero real numbers and if the system of equations a - 1 x = y + z b - 1 y = x + z c - 1 z = x + y has a non - trivial solution, then a b + b c + c a equals :

Step-by-step solution

Given, a - 1 x = y + z ⇒ a - 1 x - y - z = 0 b - 1 y = z + x ⇒ - x + b - 1 y - z = 0 c - 1 z = y + x = - x - y + c - 1 z = 0 For a non-trivial solution a - 1 - 1 - 1 - 1 b - 1 - 1 - 1 - 1 c - 1 = 0 ⇒    1 - a 1 1 1 1 - b 1 1 1 1 - c = 0 ⇒    1 - a 1 - b 1 - c - 1 - 1 1 - c - 1 + 1 - 1 - b = 0 ⇒    1 - a + b + c + a b + b c + c a - a b c - 1 + a + b + c = 0 ⇒ a b + b c + c a = a b c