Consider the following system of linear equations: x + y + z = 1 x + y + z = 1 x + y + z = It is given that the system has infinitely many solutions and < 0 . Let A be the coefficient matrix of the system and M = A^2 + 4A + 4I , where I is the identity matrix of order 3 3 . The value of ₂( ( adj ( 2 adj (M) ) ) ) is equal to :

Step-by-step solution

For the system to have infinitely many solutions, the determinant of the coefficient matrix A must be zero. (A) = vmatrix & 1 & 1 1 & & 1 1 & 1 & vmatrix = 0 Expanding the determinant: ( ^2 - 1) - 1( - 1) + 1(1 - ) = 0 ( - 1)[ ( + 1) - 1 - 1] = 0 ( - 1)( ^2 + - 2) = 0 ( - 1)^2( + 2) = 0 Since it is given that The matrix A is bmatrix -2 & 1 & 1 1 & -2 & 1 1 & 1 & -2 bmatrix . We are given M = A^2 + 4A + 4I = (A + 2I)^2 . A + 2I = bmatrix 0 & 1 & 1 1 & 0 & 1 1 & 1 & 0 bmatrix (A + 2I) = 0 - 1(0 - 1) + 1(1 - 0) = 2 Si