I am thinkng about some relations between a symmetric matrix $A_{n \times n}$ and another symmetric matrix $M_{(n+1) \times (n+1)} = \left[ \begin{array}{c |c} A & \begin{matrix} 0 \\ 0 \\-1 \\ 1 \end{matrix} \\ \hline \begin{matrix} 0& 0 & -1 & 1 \end{matrix} & 0 \end{array} \right] $.
What I have in mind about that are signature, nullity and smith normal form.
Eigenvalues of block matrix related
https://nvlpubs.nist.gov/nistpubs/jres/78B/jresv78Bn1p3_A1b.pdf
I read these two references. But I couldn't figure out yet.
I would appreciate it if you could share any ideas or comments.
Thank you in advance.