The system of linear equations has a non-trivial solution for

• Matrices Determinants

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

1. exactly one value of λ.
2. exactly two values of λ.
3. exactly three values of λ.
4. infinitely many values of λ.

Solution

For non-trivial solution,

\( \begin{bmatrix}1 & \lambda & -1 \\\lambda &-1 & -1 \\1 & 1 & -\lambda \\ \end{bmatrix}=0 \)

1(λ+1) - λ(-λ² + 1) - 1(λ + 1) = 0

λ + 1 + λ³ - λ - λ - 1 = 0

λ(λ² - 1) = 0

λ = 0, +1, -1

There are three values of λ.

The correct option is C.