Solve {c}{(1-lambda)x_{1}-2x_{2}+4x_{3}=0}{2x_{1}+(3-lambda)x_{2}+x_{3}=0}{x_{1}+x_{2}+(1-lambda)x_{3}=0}

$$\left\{ \begin{array} { c } { ( 1 - \lambda ) x _ { 1 } - 2 x _ { 2 } + 4 x _ { 3 } = 0 } \\ { 2 x _ { 1 } + ( 3 - \lambda ) x _ { 2 } + x _ { 3 } = 0 } \\ { x _ { 1 } + x _ { 2 } + ( 1 - \lambda ) x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$\left\{\begin{matrix}\\x_{1}=0\text{, }x_{2}=0\text{, }x_{3}=0\text{, }&\text{unconditionally}\\x_{1}=-\frac{2x_{3}\left(7-2\lambda \right)}{\lambda ^{2}-4\lambda +7}\text{, }x_{2}=\frac{x_{3}\left(\lambda +7\right)}{\lambda ^{2}-4\lambda +7}\text{, }x_{3}\in \mathrm{R}\text{, }&\lambda =2\text{ or }\lambda =3\text{ or }\lambda =0\end{matrix}\right.$$