Question

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.

Answer 1

λ=3,λ=1

Explanation:

let (λ-2)=a

`ax + y = 0\\x + ay = 0`

solve:

`ax + y = 0\\ax + a^2y = 0\\y-a^2y=0\\a^2 = 1`

replace a:

`(\lambda-2)^2=1\\\lambda^2-4\lambda+4=1\\\lambda^2-4\lambda+3=0`

solve:

`\lambda_1=2+√(4-3) =3\\\lambda_2=2-√(4-3)=1`