Consider the matrix
$$ B = A + \begin{bmatrix} R_{1} & R_{2} \\ R_{1} & R_{2} \end{bmatrix} = A + \begin{bmatrix} I \\ I\end{bmatrix} \begin{bmatrix} R_{1} & R_{2} \end{bmatrix} $$ where $A$ is an invertible square matrix, and $R_1$ and $R_2$ are symmetric matrices. Notice that the block matrix $R$ is akin to a rank-one matrix, in the sense that the two rows are the same.
If $R_1$ and $R_2$ were scalar, we could use the Sherman–Morrison formula to compute $B^{-1}$ in a simple way. Is there a version of that formula that would apply when $R_1$ and $R_2$ are matrices?
I think you are looking for something like the Woodbury matrix identity. In the link, choose: $U,V^{-1}$ to be eigenvectors corresponding to non-zero eigenvalues of the given matrix
$$UDU^{-1} = \begin{pmatrix} R_1 & R_2 \\ R_1 & R_2 \end{pmatrix}$$
$V = U^{-1}$, $C = D$ (in the link). Use only those columns in $U$ with non-zero eigenvalues in $D$.