Solve {r}{lambdax_{1}+x_{2}+x_{3}=0}{x_{1}+mux_{2}+x_{3}=0}{x_{1}+2mux_{2}+x_{3}=0}

$$\left. \begin{array} { r } { \lambda x _ { 1 } + x _ { 2 } + x _ { 3 } = 0 } \\ { x _ { 1 } + \mu x _ { 2 } + x _ { 3 } = 0 } \\ { x _ { 1 } + 2 \mu x _ { 2 } + x _ { 3 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3 (complex solution)

$$\left\{\begin{matrix}x_{1}=0\text{, }x_{2}=0\text{, }x_{3}=0\text{, }&\left(\lambda =0\text{ or }\mu \neq 0\right)\text{ and }\left(\lambda =\frac{1}{\mu }\text{ or }\lambda \neq 1\right)\text{ and }\left(\mu \neq 0\text{ or }\lambda \neq 1\right)\\x_{1}=-\frac{x_{3}\left(\mu -1\right)}{\lambda \mu -1}\text{, }x_{2}=-\frac{x_{3}\left(\lambda -1\right)}{\lambda \mu -1}\text{, }x_{3}\in \mathrm{C}\text{, }&\mu \neq 1\text{ and }\left(\lambda =1\text{ or }\mu =0\right)\\x_{1}=-x_{3}\text{, }x_{2}=0\text{, }x_{3}\in \mathrm{C}\text{, }&\lambda =1\text{ and }\mu =1\end{matrix}\right.$$

Solve for x_1, x_2, x_3

$$\left\{\begin{matrix}x_{1}=0\text{, }x_{2}=0\text{, }x_{3}=0\text{, }&\left(\lambda =0\text{ or }\mu \neq 0\right)\text{ and }\left(\lambda =\frac{1}{\mu }\text{ or }\lambda \neq 1\right)\text{ and }\left(\mu \neq 0\text{ or }\lambda \neq 1\right)\\x_{1}=-\frac{x_{3}\left(\mu -1\right)}{\lambda \mu -1}\text{, }x_{2}=-\frac{x_{3}\left(\lambda -1\right)}{\lambda \mu -1}\text{, }x_{3}\in \mathrm{R}\text{, }&\mu \neq 1\text{ and }\left(\lambda =1\text{ or }\mu =0\right)\\x_{1}=-x_{3}\text{, }x_{2}=0\text{, }x_{3}\in \mathrm{R}\text{, }&\lambda =1\text{ and }\mu =1\end{matrix}\right.$$