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Question

The real value of λ for which the system of equations λx + y + z = 0, -x + λy + z = 0, - x - y + λz = 0 has a non-zero solution, is _____________.

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Solution


The system of homogeneous equations λx + y + z = 0, −x + λy + z = 0 and −x − y + λz = 0 has a non-zero solution or an infinite many solutions.

=λ11-1λ1-1-1λ=0

λλ2+1-1-λ+1+11+λ=0

λ3+λ+λ-1+1+λ=0

λ3+3λ=0

λλ2+3=0

λ=02 + 3 ≠ 0 for any real value of λ)

Thus, the real value of λ for which the given system of homogeneous equations has a non-zero solution is 0.

The real value of λ for which the system of equations λx + y + z = 0, −x + λy + z = 0, −x − y + λz = 0 has a non-zero solution, is __0__.

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