Let S be the set of all values of [0, ] for which the system of linear equations x + ( )y + ( )z = 0 ( )x + y + ( )z = 0 ( )x + ( )y + z = 0 has a non-trivial solution. Then the sum of all elements of S is

Step-by-step solution

For a homogeneous system of linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero. = vmatrix 1 & & & 1 & & & 1 vmatrix = 0 Expanding the determinant: 1(1 - ^2 ) - ( - ) + ( ^2 - ) = 0 1 - ^2 - ^2 + ^2 + ^2 - ^2 = 0 (1 - ^2 ) - 2 ^2 + 2 ^2 = 0 ^2 - 2 ^2 + 2 ^2 = 0 - ^2 + 2 ^2 = 0 ^2 (2 - 1) = 0 This gives = 0 or = 1 2 . Given [0, ] : If = 0 , then = 0, . If = 1 2 , then = 3 . The sum of all possible values of is 0 + + 3 = 4 3 . Answer: 4 3