System of equations
λx+y+z=1x+λy+z=1x+y+λz=1
has no solution, so
∣λ111λ111λ∣=0
⇒∣λ+2112+λλ12+λ1λ∣=0
⇒(2+λ)∣1111λ111λ∣=0
⇒(2+λ)∣1110λ−1000λ−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