已知数列 $\left\{x_n\right\},\left\{y_n\right\},\left\{z_n\right\}$ 满足
$$
\begin{aligned}
x_0=-1, y_0 & =0, z_0=2 \text { 且 } \\
& \left\{\begin{array}{l}
x_n=-2 x_{n-1}+2 z_{n-1}, \\
y_n=-2 y_{n-1}-2 z_{n-1} \\
z_n=-6 x_{n-1}-3 y_{n-1}+3 z_{n-1}
\end{array}\right.
\end{aligned}
$$
记 $\alpha_n=\left(\begin{array}{l}x_n \\ y_n \\ z_n\end{array}\right)$ ，写出满足 $\alpha_n=A \alpha_{n-1}$ 的矩阵 $A$ ，并求 $A^n$ 及 $x_n, y_n, z_n(n=1,2, \cdots)$ 的通项表达式.