I encountered the following problem recently in a practical context.

Fix $n \ge 1$. Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form $$ X \mapsto B^{-1} (X - A) (D B^{-1} (X - A) + C)^{-1} $$ for some $A, B, C, D \in \mathbb C ^ {n \times n}$ with $B,C$ invertible.

1. What is the minimum number of pairs $(X_i, f(X_i))$ needed to determine $f$?
2. Given such a list of pairs $(X_1, f(X_1)), \dots, (X_k, f(X_k))$ and an $X$, how does one compute $f(X)$?

I fear this problem may be too simple for MathOverflow, but it lies outside my area of expertise and I will accept a reference to somewhere dealing with this sort of problem. Thank you.