Let S be the set of all ordered pairs (a, b) for which the system of linear equations x + y + z = 2 2x + 3y + az = 3 x + ay + 3z = b has infinitely many solutions. The sum of (a^2 + b^2) over all (a, b) S is

Step-by-step solution

For infinitely many solutions, = _x = _y = _z = 0 . = vmatrix 1 & 1 & 1 2 & 3 & a 1 & a & 3 vmatrix = 1(9 - a^2) - 1(6 - a) + 1(2a - 3) = -a^2 + 3a Setting = 0 a(3 - a) = 0 a = 0 or a = 3 . Case 1: a = 0 The system becomes: x + y + z = 2 2x + 3y = 3 x + 3z = b For infinitely many solutions, _x = 0 : _x = vmatrix 2 & 1 & 1 3 & 3 & 0 b & 0 & 3 vmatrix = 2(9) - 1(9) + 1(-3b) = 9 - 3b 9 - 3b = 0 b = 3 . For a = 0, b = 3 , it can be easily verified that _y = _z = 0 . So, (0, 3) S . Case 2: a = 3 The system becomes: x +