If the system of linear equations 2 x + y - z = 3 x - y - z = α 3 x + 3 y + β z = 3 has infinitely many solutions, then α + β - α β is equal to __________.

Step-by-step solution

Given, 2 x + y - z = 3 x - y - z = α 3 x + 3 y + β z = 3 Let A =   2 1 - 1 1 - 1 - 1 3 3 β     &   B = 3 α 3 For Equation to have infinitely many solutions A = 0   &   A d j .   A . B = 0 A = 2 1 - 1 1 - 1 - 1 3 3 β = 0 ⇒ - 3 β - 3 = 0 ⇒ β = - 1 Now, Minor of A = 2 β + 3 β + 3 6 β + 3 2 β + 3 3 - 2 - 1 - 3 Cofactor of A = - 2 β + 3 - β + 3 - 6 - β + 3 - 2 β + 3 - 3 2 1 3 Adjoint of A = -