Inverting a special matrix

Question

Consider matrices $A$ and $B$ of the forms below:

$$A = \lambda \cdot I$$ $$B = \beta \cdot \pmatrix{ 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots&\vdots\\ 1 & 1 & \cdots &1 }$$

In other words, $A$ is a diagonal matrix with all elements of the main diagonal equal and $B$ is a matrix of the same size as $A$ but with all of its elements equal to $\beta$.

Now, consider the matrix $C = A + B$. Is there any formula for $C^{-1}$ based on $A$, $A^{-1}$, $B$?

Thanks.

Answer 1

Probably, this paper answers to your question. See also this topic

Answer 2

I tried some examples with WolframAlpha:

It looks like:

$$A + B \in \mathbb{R}^{n\times n} \Rightarrow (A+B)^{-1} = \frac{1}{n\lambda\beta+\beta^2}\left(A-B+n\beta I\right)$$