We construct the matrix of the form
\(B=\begin{pmatrix}
0 & -1 & 1 \\
-1 & 1 & 0 \\
1 & 0 & -1
\end{pmatrix}\)
Perform the Gaussian elimination by making each transformation also on the corresponding column:
Swap the first and second rows and then the first and second columns:
\(
\begin{pmatrix}
0 & -1 & 1 \\
-1 & 1 & 0 \\
1 & 0 & -1
\end{pmatrix}
\sim
\begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
1 & 0 & -1
\end{pmatrix}
\sim
\begin{pmatrix}
1 & -1 & 0 \\
-1& 0 & 1 \\
0 & 1 & -1
\end{pmatrix}
\)
Add the first row to the second row and then also for the columns:
\(
\begin{pmatrix}
1 & -1 & 0 \\
-1& 0 & 1 \\
0 & 1 & -1
\end{pmatrix}
\sim
\begin{pmatrix}
1 & -1 & 0 \\
0 & -1 & 1 \\
0 & 1 & -1
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 1 \\
0 & 1 & -1
\end{pmatrix}
\)
Add the second row to the third row and then also for the columns:
\(
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 1 \\
0 & 1 & -1
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 1 \\
0 & 0 & 0
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{pmatrix}
\)
The signature is a triple consisting of the numbers of positive, negative and zero values on the diagonal of the resulting diagonal the matrix.
