The sum of three numbers is 20 . If we multiply the first number by 2 and add the second number to the result, and subtract the third number, we get 23 . By adding second and third numbers to three times the first number, we get 46. The above situation can be represented in matrix form as A X=B then the value of determinant formed by the cofactors of matrix A is

Step-by-step solution

Let the three numbers be x, y, z . The given conditions are: x + y + z = 20 2x + y - z = 23 3x + y + z = 46 The matrix A is given by: A = pmatrix 1 & 1 & 1 2 & 1 & -1 3 & 1 & 1 pmatrix The determinant of A is |A| = 1(1 - (-1)) - 1(2 - (-3)) + 1(2 - 3) = 1(2) - 1(5) + 1(-1) = 2 - 5 - 1 = -4 . Let C be the matrix of cofactors of A . The determinant of the cofactor matrix is given by |C| = |A|^ n-1 , where n is the order of the matrix. Here n = 3 . |C| = |A|³⁻¹ = |A|^2 . Substituting |A| = -4 , we get |C| = (-4)^2 = 1