The symmetric matrix I need to invert is of the following form:

\begin{align} J_e=\left(\begin{matrix}-I&B_{11}&B_{12}&...&...&B_{1(N-1)}\\ B_{11}&-I&B_{22}&...&...&B_{2(N-1)}\\ \vdots&&\ddots&&&\vdots\\ B_{i1}&&&-I&...&B_{i(N-1)}\\ \vdots&...&B_{ij}&...&\ddots&\vdots\\ B_{(N-1)1}&...&...&...&...&-I\end{matrix}\right) \end{align}

where $I=I_{N-1}$ for some $N\geq3$ and $B_{ij}=B_{ji}'$ are $(N-1)\times(N-1)$ matrices with only one non-zero entry $[B_{ij}]_{ij}\neq0$, thus they all have rank one. The entire matrix is thus $N(N-1)\times N(N-1)$.

**My question: How can I analytically obtain $J_e^{-1}$ exploiting the above structure?**

I was thinking about writing it as \begin{align} J_e=M-I=\left(\sum_{k=1}^{N-1}M_k\right)-I \end{align}

where each $M_k$ is a monomial matrix, but the requirements of the results I found so far that relate to this are always to demanding for what the above problem.

Any hint or idea is very much appreciated.