T \left( \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right) = \left( \begin{array} { c } { x + y } \\ { 3 x - 2 y } \\ { x - y + z } \end{array} \right)
Solve for x, y, z (complex solution)
\left\{\begin{matrix}x=0\text{, }y=0\text{, }z=0\text{, }&T\neq 1\\x=0\text{, }y=0\text{, }z\in \mathrm{C}\text{, }&T=1\\x=\frac{\left(3-\sqrt{21}\right)y}{6}\text{, }y=3z\text{, }z\in \mathrm{C}\text{, }&T=\frac{-\sqrt{21}-1}{2}\\x=\frac{\left(\sqrt{21}+3\right)y}{6}\text{, }y=3z\text{, }z\in \mathrm{C}\text{, }&T=\frac{\sqrt{21}-1}{2}\end{matrix}\right.
Solve for x, y, z
\left\{\begin{matrix}x=0\text{, }y=0\text{, }z=0\text{, }&T\neq \frac{\sqrt{21}-1}{2}\text{ and }T\neq \frac{-\sqrt{21}-1}{2}\text{ and }T\neq 1\\x=0\text{, }y=0\text{, }z\in \mathrm{R}\text{, }&T=1\\x=\frac{\left(3-\sqrt{21}\right)y}{6}\text{, }y=3z\text{, }z\in \mathrm{R}\text{, }&T=\frac{-\sqrt{21}-1}{2}\\x=\frac{\left(\sqrt{21}+3\right)y}{6}\text{, }y=3z\text{, }z\in \mathrm{R}\text{, }&T=\frac{\sqrt{21}-1}{2}\end{matrix}\right.
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