For a real number α , if the system 1 α α 2 α 1 α α 2 α 1   x y z = 1 - 1 1 of linear equations, has infinitely many solutions, then 1 + α + α 2 =  

Step-by-step solution

Determinant value of 3 × 3 matrix should be zero for infinite solution Δ = 0 ⇒ 1 1 - α 2 - α α - α 3 + α 2 α 2 - α 2 = 0 1 - α 2 - α 2 + α 4 = 0 α 2 - 1 2 = 0 ⇒ α = ± 1 But at α = 1 , three equations become parallel, so no solution so rejected at α = - 1 , All three equation become x - y + z = 1 (coincident planes), so infinite solution. ∴ 1 + α + α 2 = 1