If the system of linear equations x + y + z = 2 2x + 3y + z = 5 3x + 4y + 2z = has infinitely many solutions, and A = pmatrix 2 & & 3 & 1 & 2 1 & 2 & pmatrix , then the value of |adj(A)| is equal to :

Step-by-step solution

For the system to have infinitely many solutions, we must have = 0 and ₁ = ₂ = ₃ = 0 . = vmatrix 1 & 1 & 1 2 & 3 & 3 & 4 & 2 vmatrix = 1(6 - 4 ) - 1(4 - 3 ) + 1(8 - 9) = 1 - . Setting = 0 gives = 1 . Now, evaluate ₃ : ₃ = vmatrix 1 & 1 & 2 2 & 3 & 5 3 & 4 & vmatrix = 1(3 - 20) - 1(2 - 15) + 2(8 - 9) = - 7 . Setting ₃ = 0 gives = 7 . Substitute = 1 and = 7 into matrix A : A = pmatrix 2 & 1 & 3 7 & 1 & 2 1 & 2 & 1 pmatrix . The determinant of A is: |A| = 2(1 - 4) - 1(7 - 2) + 3(14 - 1) = 2(-3) - 5 + 3(13) = -6 - 5 +