Suppose I have a symmetric matrix $A_{n\times n}$ and $A^{-1}_{n\times n}$ is its inverse. Then I randomly exclude the $i$-th row and column, where $1\le i\le n \in \mathbb{N}$, obtaining the symmetric matrix $A_{n-1\times n-1}$.
Is there any way to calculate the inverse of $A_{n-1\times n-1}$ using the inverse already calculated $A^{-1}_{n\times n}$ to simplify the calculus?
Example : let $n=3$ and consider
$A=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}.$
If you exclude the first row and the first column you will obtain a $ 2 \times 2 -$ matrix which is not invertble !