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For which values of λ does the system of equations (λ − 2)x + y = 0 x + (λ − 2)y = 0 have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.

Respuesta :

Chenk13

Answer:

λ=3,λ=1

Step-by-step explanation:

let (λ-2)=a

[tex]ax + y = 0\\x + ay = 0[/tex]

solve:

[tex]ax + y = 0\\ax + a^2y = 0\\y-a^2y=0\\a^2 = 1[/tex]

replace a:

[tex](\lambda-2)^2=1\\\lambda^2-4\lambda+4=1\\\lambda^2-4\lambda+3=0[/tex]

solve:

[tex]\lambda_1=2+\sqrt{4-3} =3\\\lambda_2=2-\sqrt{4-3}=1[/tex]

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