Suppose I have a discrete state space model:
\begin{align} \theta[k+1] &= A \theta[k] + B u[k]\\ y[k] &= C \theta[k] \end{align}
I know that the equivalent transfer function can be found by solving
$$ Y(z) = C (zI-A)^{-1} B U(z) $$
But is there a quick way to determine the order of the transfer function for each input? I.e., the degree of its numerator and denominators, assuming that the dimensions of $A$ will be $n\times n$, $B$'s will be $n \times m$ and $C$'s will be $1 \times n$? (Assuming here there is only one variable being observed.)