Sum of inverse and inverse adjoint

Let $A=\begin{pmatrix}a&0\ c&d\end{pmatrix}$ be a real matrix, where $ad=1$ and $c\neq 0$. If $$A^{-1}+(\mathrm{adj}\,A)^{-1}=\begin{pmatrix}\alpha&\beta\ \gamma&\delta\end{pmatrix},$$ then $(\alpha,\beta,\gamma,\delta)$ is equal to