If the system of linear equations x - y + 2 z = 3 2 x + y - z = 4 7 x - y + λ z = μ has infinitely many solutions, and a matrix M is defined as M = 2 λ - 1 μ , then the value of | a d j ( M ) | is

Step-by-step solution

For the system to have infinitely many solutions, the third equation must be a linear combination of the first two equations. Let the third equation be a ( E q 1 ) + b ( E q 2 ) . Comparing the coefficients of x and y : a + 2 b = 7 - a + b = - 1 Adding these two equations gives 3 b = 6 ⇒ b = 2 . Substituting b = 2 into the second equation gives - a + 2 = - 1 ⇒ a = 3 . Now, comparing the coefficients of z and the constant terms: λ = 2 a - b = 2 ( 3 ) - 2 = 4 μ = 3 a + 4 b = 3 ( 3 ) + 4 ( 2 ) =