suppose I have an asymptotic expansion for a matrix-valued function $\psi : \mathbb{C} \to \mathbb{C}^{2 \times 2}$ : $$\psi(\lambda) \sim I + \frac{m_1}{\lambda} + \frac{m_2}{\lambda^2} + \cdots \ \ \ \ \lambda \to \infty$$ where $m_i$ are constant $2 \times 2$ matrices. we also know that $\psi(\lambda)$ is invertible for all $\lambda$. My question is : Given $m_is$, How can I write an asymptotic expansion for $\psi^{-1}(\lambda)$ as $\lambda \to \infty$, I need the first $4$ terms of this expansion.
Any help is really appreciated, Thanks