Find the real values of λ for which the followings system of linear equations has non – trivial solutions. Also, find the non – trivial solutions

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Find the real values of λ for which the followings system of linear equations has non – trivial solutions.

Also,

find the non – trivial solutions

2λx – 2y + 3z = 0

x + λy + 2z = 0

2x + λz = 0

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Given Equations :

2λx – 2y + 3z = 0

x + λy + 2z = 0

2x + λz = 0

For trivial solution D = 0

= 2λ (λ×λ – 0×2) + 2(1×λ – 2×2) + 3(1×0 – 2×λ)

= 2λ (λ2 – 0) + 2(λ – 4) + 3(0 – 2λ)

= 2λ 3 + 2λ – 8 – 6λ

= 2λ 3 + 4λ – 8

Now,

D = 0

3 – 4λ – 8 = 0

3 – 4λ= 8

λ(λ2 – 2) = 4

Hence,

λ = 2

Now,

let z = k

⇒ 4x – 2y = – 3k

And x + 2y = – 2k

Now using the cramer’s rule