An ordered pair (α,  β)for which the system of linear equations (1+α) x+βy+z=2,αx+(1+β)  y+z=3,    αx+βy+2z=2 has a unique solution is

The condition for a non-homogeneous system to have a unique solution is Δ≠0⇒1+αβ1α1+β1αβ2≠0⇒R2→R2−R3R1→R1−R3, 10-101-1αβ2   ≠ 0⇒α+β+2≠0⇒(α,  β)=(2,  4)is the only possible answer