inverse of the sum of a diagonal and a symmetric matrix

Question

I need to compute the inverse of the following sum of matrices:

$$\begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix} +D $$

where B is a non-negative matrix and D is a non-negative diagonal matrix. They are both real an square matrices.

Answer 1

By definition, non-negative matrices are those matrices with non-negative entries. Suppose, $\begin{pmatrix} 0&2\\2&0\end{pmatrix}=\begin{pmatrix} 0&B\\B^{T}&0\end{pmatrix}$ and $D=\begin{pmatrix} 2&0\\0&2\end{pmatrix}$. Then the inverse of the sum of matrices doesn't exist.