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The system of linear equations
x + λy - z = 0
λx - y - z = 0
x + y - λz = 0
has a non-trivial solution for
- infinitely many values of λ
- exactly one value of λ
- exactly two values of λ
- exactly three values of λ
Correct answer: exactly three values of λ
Solution
The system has a non-trivial solution when the determinant of the coefficient matrix is zero, which occurs for specific values of λ. In this case, the determinant is a polynomial in λ that can be factored to find three distinct roots, indicating that there are exactly three values of λ for which the system has non-trivial solutions.
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