Consider the system of linear equations: x + ay + z = 1 ax + y + z = 2 x + y + az = b The complete set of conditions on a and b for which the system has no solution is

Step-by-step solution

For the system to have no solution, the determinant of the coefficient matrix must be zero: = vmatrix 1 & a & 1 a & 1 & 1 1 & 1 & a vmatrix = 0 Expanding the determinant: 1(a - 1) - a(a^2 - 1) + 1(a - 1) = 0 (a - 1)[1 - a(a + 1) + 1] = 0 (a - 1)(2 - a - a^2) = 0 -(a - 1)^2(a + 2) = 0 a = 1 or a = -2 Case 1: a = 1 The first two equations become x + y + z = 1 and x + y + z = 2 . These represent parallel, distinct planes and are inherently contradictory. Thus, the system has no solution for any real value of b . Condi