Let a , b and c be any real numbers. Suppose that there are real numbers x , y and z not all zero such that x = c y + b z , y = a z + c x and z = b x + a y , then a 2 + b 2 + c 2 + 2 a b c is equal to

Step-by-step solution

System of equations x - c y - b z = 0 c x - y + a z = 0 b x + a y - z = 0 has non trivial solution if the determinant of coefficient matrix is zero. ⇒ 1 - c - b c - 1 a b a - 1 = 0 ⇒ 1 1 - a 2 + c - c - a b - b c a + b = 0 ⇒ a 2 + b 2 + c 2 + 2 a b c = 1