(Solved): Determine the values of \( \lambda \) for which the following system of equations has nontrivial s ...

Determine the values of \( \lambda \) for which the following system of equations has nontrivial solutions. Find the solutions for each value of \( \lambda \). \[ \begin{array}{l} (5-\lambda) x_{1}+4 x_{2}+2 x_{3}=0 \\ 4 x_{1}+(5-\lambda) x_{2}+2 x_{3}=0 \\ 2 x_{1}+2 x_{2}+(2-\lambda) x_{3}=0 \end{array} \]