(2024,1)已知数列x_{n)},y_{n)},z_{n)}满足x_(0)=-1,y_(0)=0,z_(0)=2,且}x_(n)=-2x_(n-1)+2z_(n-1),y_(n)=-2y_(n-1)-2z_(n-1),z_(n)=-6x_(n-1)-3y_(n-1)+3z_(n-1).记alpha_(n)=}x_(n)y_(n)z_(n)(n=1,2,···).

1. **确定矩阵 $A$**： 根据递推关系，得 \[ A = \begin{pmatrix} -2 & 0 & 2 \\ 0 & -2 & -2 \\ -6 & -3 & 3 \end{pmatrix}. \] 2. **求特征值和特征向量**： 特征值为 $\lambda_1 = -2$, $\lambda_2 = 0$, $\lambda_3 = 1$。 3. **对角化**： 令 $P = \begin{pmatrix} 1 & 1 & 2 \\ -2 & -1 & -2 \\ 0 & 1 & 3 \end{pmatrix}$，计算 $P^{-1}$ 后得 \[ A^n = PD^nP^{-1}. \] 4. **答案**： \[ \boxed{ \begin{cases} x_n = \frac{1}{3} \left[ 3^n - (-2)^n \right], \\ y_n = \frac{2}{3} \left[ (-2)^n - 3^n \right], \\ z_n = 3^n - \frac{2}{3} (-2)^n \end{cases} \] 5. **总结**： �{2024,1) \boxed{ \boxed{ \begin{cases} x_n = \frac{1}{3} \left[ 3^n - (-2)^n \right], \\ y_n = \frac{2}{3} \left[ (-2)^n - 3^n \right], \\ z_n = 3^n - \frac{2}{3} (-2)^n \end{cases} \] 6. ** \[ \boxed{ \boxed{ x_n = \frac{1}{3} \left[ 3^n - (-2)^n \right] \] \] \[ \boxed{ \] \[ \boxed{ \boxed{ \boxed{ \boxed{ \] \[ \boxed{ \[ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \] \] \boxed{ \boxed{ \boxed{ \] \] \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \boxed} \] \[ \boxed{ \] \[ \boxed{ \[ \boxed{ \] \[ \boxed{ \] \[ \boxed{ \boxed{ \] \[ \boxed{ \boxed{ \] \[ \boxed{ \] \[ \[ \boxed{ \] \[ \boxed{ \] \[ \boxed{ \boxed{ \boxed{ \] \] \[ \boxed{ \boxed{ \] \[ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \boxed{ \] \boxed{ \[ \boxed{ \] \[ \boxed{ \boxed{ \] 6. \[ \[ \boxed{ \] \[ \] \text。 \[ \[ \] \[ \[ \[ \] \[ \[ \] \[ \[ \[ \[ \[ \[ \[ \] \[ \] \[ \[ \[ \[ \[ \[ \[ \] \[ \ \ ** \[ \[ \[ \[ \2 \**: \[ \2\4 \ \ \ \ 2 \6 ** \[ \ \[ \[ \2 \[ \ \ ** \[ \22 \ \0 \ ** \0 \[ \2 \4 \ \ \ \ \ \ \ \ \ \ **\3 \ ** \0 \0 \ \ ** \ \0 \ ** \0 \2 \4 \ \ ** \ \4. \ \ ** \2 \3 \33 \ \2 \ \41 \ \ \ \ \ \ \ \ \ \ \ \ \12 \3 \ \ \ ** \ \ \ \ \2 \\2 \ \3 2 \ \ \ \ \ \00{ ** \4 \ \ \23 \ \44 \ ** \ \43 \3 \33 \ \ \ \3 \00 \20 \ \2 \4\33 \43 \ \12\ \4\ \0 \1 \0 \ \22 \44 \33 ** \ \