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Question

Find tbe sum of all the values of λ for which the system of equations λx+y+z=1, x+λy+z=λ, x+y+λz=λ2 has infinite number of solutions.

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Solution

letAbethecoefficientmatrix,|A|=∣ ∣λ111λ111λ∣ ∣=(λ1)2(λ+2)ifΔ=0,Δ1,Δ2,Δ30inconsistentΔ=0,anyofΔ1,Δ2,Δ3iszeroinfinitesolutionsforλ=1Δ=0,allΔ1,Δ2,Δ3=0λ=2Δ=0,Δ1,Δ2,Δ30Henceforλ=1givensystemhasinfinitenoofsolutions.

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