System of nonhomogeneous difference equation

[smilieevah]

How do you solve the system z(t+1)=Az(t)+b where A is a 2x2 matrix and z(t+1), z(t), b are 2x1 matricies?
I solved the homogeneous solution: z(t)=P(D^t)(P^-1)z(0) where D is the diagonal matrix of eigenvalues of A and P is the matrix of eigenvectors.
I tried to solve the nonhomogeneous solution at the steady state where z(t+1)=z(t). I'm not sure if this is the right method.
Then I added the two solutions z(t) for the homogeneous and the nonhomogeneous equations.

 

[defunc]

Guess a constant vector for the particular solution. This will give
$$x_{p}=(I-A)^{-1}b$$
Then add it to the homogeneous solution to obtain the solution.