A matrix valued function is of the form $\psi:\mathbb{R}_+\to\mathbb{R}^{n\times n}$ and it is known that $\psi(\lambda)$ is always a **positive definite** matrix. The asymptotic exapnsion of $\psi(\lambda)$ is given as $$\psi(\lambda) = A + \frac{B_1}{\lambda}+ \frac{B_2}{\lambda^2}+ \frac{B_3}{\lambda^3}+...$$ where $A$ is an $n\times n$ **all ones** matrix and $B_1,B_2,B_3...$ are nonsingular $n\times n$ matrices.

I want to get an asymptotic expansion for $\psi(\lambda)^{-1}$. I don't want to compute the exact coefficient matrices but want to show that it is of the form $$\psi(\lambda)^{-1} = \lambda D_{-1} + D_0 + \frac{D_1}{\lambda} + \frac{D_2}{\lambda^2} + \frac{D_3}{\lambda^3} +...$$ where $D_{-1},D_0,D_1,D_2,..$ are $n\times n$ matrices and $D_{-1}$ is a non zero matrix.

What I know

Neumann's series and this post.

PS : if this isn't a research level problem, please let me know.