Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a constant. We can assume $M = 0$ for the time being. Note that matrices $A$ and $C$ may be singular.
While the dimensions and values of the matrices can be arbitrary, here's an example for the $3 \times 3$ case.
$$ A = \begin{bmatrix} 2 & \alpha & 3\\ 0 & 11 & 5\\ -1 & \beta & 0 \end{bmatrix}, \qquad C = \begin{bmatrix} 4 & 3 & 7\\ 2.2 & 9 & 5\\ 2 & 0.2 & -1 \end{bmatrix}, \qquad X_t = \begin{bmatrix} 4\\ 2\\ 3 \end{bmatrix} $$
Imagine we have found a matrix $B$ that gives us a solution for $X_{t+1}$ of the form
$$ X_{t+1} = BX_t$$
($B$ may have been computed numerically, if an analytic expression is difficult). We therefore have
$$ ABX_t = CX_t $$ and $$ AB = C $$
The Question
Say we perturb one element of $A$ (e.g. $A \rightarrow A + \Delta_{kl}$, with $\Delta_{kl}$ being a matrix of zeros for all elements except for element $k,l$ with takes a small positive value $\delta$), what effect does this have on the value of an element $B_{ij}$ of matrix $B$?
If $(k,l)=(1,2)$ this is basically asking for $ \partial B_{ij} / \partial \alpha$ in the example above.
Is it possible to derive an expression for this quantity? (and not just use a numerical solver)
