Let $A, A^{-1} \in \mathbb{R}^{n \times n}$ be matrices. Both $A$ and $A^{-1}$ are known. Suppose we have a Matrix $B \in \mathbb{R}^{n+1 \times n+1}$ of following form:

\begin{bmatrix} A& b\\ c& 1 \end{bmatrix}

where $b$ is a column vector, $c$ is a row vector and $B$ is invertible. How can I calculate $B^{-1}$ with known $A, A^{-1}$? Can the formula of Sherman–Morrison be applied here? If yes, how?

As far as I understand, it can be applied if some perturbation is made to $A$. But the problem here is that $B$ has a different shape than $A$. Appending $A$ with zero entries in the beginning will not work either because the same matrix with zeros added in the bottom row and the right column is not necessarily nonsingular.