The above equation is a nonhomogeneous difference equation. To find the general solution, I have to find the particular solution and a complementary solution.
I have obtained the complementary solution, that is, for the homogeneous difference equation $z_{t+1} = A z_{t}$, which is $$z_{c} = c_{1}\left ( -2 \right )^{n}\begin{bmatrix} 2\\ 1 \end{bmatrix} + c_{2}\left ( 3 \right )^{n}\begin{bmatrix} 1\\ 2 \end{bmatrix}.$$
This is after calculating the eigenvalues and the eigenvalues for the matrix $A$.
Now for finding the particular solution, $z_{p}$ I was wondering if I can get some help in guessing a function.
The general solution for the nonhomogeneous difference equation will consist of $z_{n} = z_{c} + z_{p}$.
EDIT: For the particular solution, I have solved $z^{*} = A z^{*} + b$ and have solved for vector $$z^{*} = \begin{bmatrix} -1\\ -11 \end{bmatrix} = z_{p}.$$
Thus, for the general solution we have, $$z_{n} = c_{1}\left ( -2 \right )^{n}\begin{bmatrix} 2\\ 1 \end{bmatrix} + c_{2}\left ( 3 \right )^{n}\begin{bmatrix} 1\\ 2 \end{bmatrix} + \begin{bmatrix} -1\\ -11 \end{bmatrix}.$$
Does this general solution for the nonhomogeneous difference equation make sense?