Rewriting the equations,
x(−1−λ)+y=0
3x+y(1−λ)=0
(1) [(−1−λ)31(1−λ)][xy]=[00]
(−1−λ)(1−λ)−3=−1+λ−λ+λ2−3
=λ2−4
(2) when det(λI2−A)=0
⇒λ2−4=0
⇒λ=±2
(3) Substituting the value of (λI2−A)X=0
For system to be consistent, λ=±2
(λI2−A)X=0
[(−1−λ)31(1−λ)][xy]=[00]
x(−1−λ)+y=0
x(−1−2)+y=0
−3x+y=0
3x−y=0
3x+y(1−λ)=0
3x+y(1−2)=0
3x−y=0
Since both the equations are parallel therefore the system has infinitely many solutions.
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