I want to calculate inverse of the following matrix$$I_a=\sum \limits _{j=1}^J(n_{a,j}-n_{a,j+1})\begin{pmatrix}\Sigma _j^{-1} & 0_{j\times (J-j)} \\ 0_{(J-j)\times j} & 0_{(J-j)\times (J-j)}\end{pmatrix},$$where $\Sigma _j$ is the leading $j\times j$ sub matrix of $\Sigma _{J\times J}$, $n_j$ and $n_{j+1}$ are scalars and $n_{J+1}=0$.
I wonder if there is any way to get an explict form of $[I^{-1}]_{JJ}$, i.e. $J$th diagonal element of $I^{-1}$. Also, $\Sigma _{J\times J}$ is a symmetric positive semi definite matrix (covariance matrix). I tried to use decompositions but did not help. I appreciate any help in this problem.