The sum of three numbers is 6 . Twice the third number, when added to the first number gives 7 , On adding the sum of the second and third numbers to thrice the first number, we get 12 . The above situation can be represented in matrix form as A X=B . Then the | adj A| is equal to

Step-by-step solution

Let the three numbers be x, y, z . The given conditions are: 1. x + y + z = 6 2. x + 0y + 2z = 7 3. 3x + y + z = 12 The system of equations in matrix form AX = B is: pmatrix 1 & 1 & 1 1 & 0 & 2 3 & 1 & 1 pmatrix pmatrix x y z pmatrix = pmatrix 6 7 12 pmatrix Calculate the determinant of matrix A : |A| = 1(0 - 2) - 1(1 - 6) + 1(1 - 0) |A| = -2 + 5 + 1 = 4 The property of the adjoint matrix is | adj A| = |A|^ n-1 , where n is the order of the matrix. Here n = 3 , so | adj A| = |A|³⁻¹ = |A|^2 . | adj A| = 4^2 = 16 . A