Solve Matrix

$$\left. \begin{array} { l } { \text { If } A = \begin{bmatrix} \begin{array} { c c } { 3 } & { - 2 } \\ { \lambda } & { - 2 } \end{array} \end{bmatrix} \text { , find the value } } \\ { \text { of } \lambda \text { so that } A ^ { 2 } = \lambda A - 2 I \text { . } } \end{array} \right.$$

Solve for A, λ, I

$$\left\{\begin{matrix}\\A=\frac{\sqrt{\lambda ^{2}-8I}+\lambda }{2}\text{, }\lambda \in \mathrm{C}\text{, }I\neq 0\text{; }A=\frac{-\sqrt{\lambda ^{2}-8I}+\lambda }{2}\text{, }\lambda \in \mathrm{C}\text{, }I\neq 0\text{; }A=0\text{, }\lambda =0\text{, }I=0\text{, }&\text{unconditionally}\\A=\frac{\sqrt{\lambda ^{2}-8I}+\lambda }{2}\text{, }\lambda \in \mathrm{C}\text{, }I\in \mathrm{C}\text{, }&arg(\lambda )<\pi \text{ and }\lambda \neq 0\\A=0\text{, }\lambda \in \mathrm{C}\text{, }I=0\text{, }&\lambda \neq 0\\A=\frac{-\sqrt{\lambda ^{2}-8I}+\lambda }{2}\text{, }\lambda \in \mathrm{C}\text{, }I\in \mathrm{C}\text{, }&|\frac{arg(\lambda ^{2})}{2}-arg(\lambda )|\geq \pi \text{ and }\lambda \neq 0\end{matrix}\right.$$