Let S denote the set of all real values of λ such that the system of equations λ x + y + z = 1 x + λ y + z = 1 x + y + λ z = 1 is inconsistent, then ∑ λ ∈ S λ 2 + λ is equal to

Step-by-step solution

System of equations λ x + y + z = 1 x + λ y + z = 1 x + y + λ z = 1 has no solution, so λ 1 1 1 λ 1 1 1 λ = 0 ⇒ λ + 2 2 + λ 2 + λ 1 λ 1 1 1 λ = 0 ⇒ 2 + λ 1 1 1 1 λ 1 1 1 λ = 0 ⇒ 2 + λ 1 0 0 1 λ - 1 0 1 0 λ - 1 = 0 ⇒ 2 + λ λ - 1 2 = 0 ⇒ λ = 1 ,   - 2 For λ = 1 , we get infinite solutions, since equation will be identical. So, ∑ λ 2 + λ = 4 + 2 = 6