Let $\bf A$ be an $n \times n$ non-singular  matrix over $\mathbb{F}$. 
Let $x$ be a non-zero element of $\mathbb{F}$.
Suppose that  ${\bf 1}_{n}$ is a symbol for  the all-one  vector  of length $n$ over $\mathbb{F}$. 
Now consider the following  $(n+1) \times (n+1)$ matrix and assume that $\bf B$ is a invertible matrix over $\mathbb{F}$. 
$$
{\bf B}=
\left(
 \begin{array}{cc}
x &{\bf 1}^T_{n} \\
 {\bf 1}_{n} & {\bf A}
 \end{array}
 \right).
 $$ 

My question: Is there a closed-form expression for the inverse of $\bf B$, denoted with ${\bf B}^{-1}$?


Thanks for any help.