Let the homogeneous system of linear equations ( )x - y + z = 0 ( )x + 4y + 3z = 0 2x + 7y + 7z = 0 have non-trivial solutions. The number of values of in the interval [0, 2 ] is

Step-by-step solution

For a homogeneous system to have non-trivial solutions, the determinant of the coefficient matrix must be zero. = vmatrix & -1 & 1 & 4 & 3 2 & 7 & 7 vmatrix = 0 Expanding the determinant: (28 - 21) - (-1)(7 - 6) + 1(7 - 8) = 0 7 + 7 - 6 + 7 - 8 = 0 7 + 14 - 14 = 0 + 2 = 2 Rearranging gives = 2(1 - ) . Squaring both sides: ^2 = 4(1 - 2 + ^2 ) 1 - ^2 = 4 - 8 + 4 ^2 5 ^2 - 8 + 3 = 0 Factoring the quadratic equation: (5 - 3)( - 1) = 0 Case 1: = 1 This gives = 0 and = 2 . Substituting back into + 2 = 2 yields = 0 , whic