Inverse of a particular matrix: $A = D + aJ$, all entries of $J$ are equal to $1$

Question

Let $$A = D + aJ$$ where $D = \mbox{diag}(d_1,\ldots,d_{n})$ and $J$ is $n\times n$ matrix of all ones. Is it possible to find $A^{-1}$ analytically?

Answer 1

If $d_1, d_2, \dots, d_n \neq 0$, then, using the Sherman-Morrison formula,

$$\left( \mathrm D + a 1_n 1_n^\top \right)^{-1} = \mathrm D^{-1} - a \frac{\mathrm D^{-1} 1_n 1_n^\top \mathrm D^{-1}}{1 + a 1_n^\top \mathrm D^{-1} 1_n}$$