Let the system of linear equations 2x - y + 3z = 5 x + 2y - z = 4x + 3y + z = 11 have infinitely many solutions. If a matrix M is defined as M = bmatrix 1 & & & & 1 & 1 & bmatrix , then the value of | adj (M)| is equal to

Step-by-step solution

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. = vmatrix 2 & -1 & 3 1 & 2 & -1 4 & 3 & vmatrix = 0 Expanding along the first row: 2(2 + 3) - (-1)( + 4) + 3(3 - 8) = 0 4 + 6 + + 4 - 15 = 0 5 - 5 = 0 = 1 For the system to be consistent and have infinitely many solutions, the third equation must be a linear combination of the first two. Let R₃ = c₁R₁ + c₂R₂ . Comparing coefficients of x and y : 2c₁ + c₂ = 4 -c₁ + 2c₂ = 3 From the first equation, c₂ = 4 - 2c₁ .