Big-O / smal-o for perturbed matrix-inversion: What is $(A_n + \mathcal O_n(n^{-\alpha}))^{-1}$?

Question

Let $\alpha > 0$ and for each natural number, let $B_n$ be a square matrix $B_n = \mathcal O(n^{-\alpha})$ as $n \rightarrow \infty$. Suppose $A_n$ is ineverible for every $n$ and $A_n \rightarrow A$ (invertible).

Question

What is the order of $(A_n + B_n)^{-1}$ ?