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Question:

The system of linear equations x + λy - z = 0, λx - y - z = 0, x + y - λz = 0 has a non-trivial solution for:

exactly one value of λ

exactly two values of λ

infinitely many values of λ

exactly three values of λ

Solution:

|1 λ -1|
|λ -1 -1|=0
|1 1 -λ|
R3 = R3 - R1
|1 λ -1|
|λ -1 -1|=0
|0 1-λ -λ+1|
R2 = R2 - R1
|1 λ -1|
|0 1-λ -λ+1|=0
|0 1-λ λ-1|
|1 λ -1|
|0 1-λ -λ+1|=0
Expand it ⇒ -λ(λ² - 1) = 0
λ = 0, 1, -1