Let $A, A^{-1} \in \mathbb{R}^{n \times n}$ be known matrices. Suppose we have an invertible matrix $B \in \mathbb{R}^{(n+1) \times (n+1)}$ of the following form:

$$B = \begin{bmatrix} A & b\\ b^T & 1 \end{bmatrix}$$

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

As far as I understand, it can be applied if some perturbation is made to $A$. However, 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.