If the system of linear equations aligned & x + 2y + z = 3 & 3x + y - 2z = 4 & -x + 3y + az = b aligned has infinitely many solutions, and M is a matrix defined as M = bmatrix a & 1 & 1 1 & b & 1 1 & 1 & 1 bmatrix , then the value of | adj (M)| is equal to :

Step-by-step solution

For the given system of equations to have infinitely many solutions, we must have D = D₁ = D₂ = D₃ = 0 . D = vmatrix 1 & 2 & 1 3 & 1 & -2 -1 & 3 & a vmatrix = 1(a + 6) - 2(3a - 2) + 1(9 + 1) = -5a + 20 Setting D = 0 , we get a = 4 . Now, D₃ = vmatrix 1 & 2 & 3 3 & 1 & 4 -1 & 3 & b vmatrix = 1(b - 12) - 2(3b + 4) + 3(9 + 1) = -5b + 10 Setting D₃ = 0 , we get b = 2 . Substitute a = 4 and b = 2 into the matrix M : M = bmatrix 4 & 1 & 1 1 & 2 & 1 1 & 1 & 1 bmatrix The determinant of M is: |M| = 4(2 - 1) - 1(1 - 1) + 1(