Find all values of 𝜆

Question

Find all values of 𝜆 (the Greek letter lambda) for which the homogeneous linear system has nontrivial solutions. (Enter your answers as a comma-separated list.)

(𝜆-4)x + y = 0

x+(𝜆-4)y = 0

Answer 1

(λ - 4)x + y = 0

So, y = -(λ - 4)x.

Therefore, x - (λ - 4)²x = 0

[1 - (λ - 4)²]x = 0

If x = 0, then, from the first equation y = 0 also. So, we get the trivial solution (0,0).

So, we can assume that x is nonzero.

Therefore, (λ - 4)² = 1.

So, λ² - 8λ - 17 = 0

λ = [8 ± √132] / 2 = 4 ± √33