Form signature with parameter

In dependence on the parameter \(a\in\mathbb R\) determine the signature of the form with the matrix

Variant
\( \begin{pmatrix} 1 &1 &1\\ 1 &0 &0\\ 1 &0 &a \end{pmatrix} \)
Solution
We perform Gaussian elimination by applying each transformation also on the corresponding column:

Subtract the first row/column from the others and then the second from the third:
\( \begin{pmatrix} 1 &1 &1\\ 1 &0 &0\\ 1 &0 &a \end{pmatrix} \sim\sim \begin{pmatrix} 1 &0 &0\\ 0 &-1 &-1\\ 0 &-1 &a-1 \end{pmatrix} \sim\sim \begin{pmatrix} 1 &0 &0\\ 0 &-1 &0\\ 0 &0 &a \end{pmatrix} \)
The signatutre depends on the sign of \(a\), as this will affect the number of positive, negative or zero elements of the resulting diagonal matrix.
Answer
For \(a\gt 0\) the signature is \((2{,}1,0)\), for \(a\lt 0\) the signature is \((1{,}2,0)\), and for \(a=0\) the signature is \((1{,}1,1)\).
Variant
\( \begin{pmatrix} 1 &2 &3\\ 2 &a &0\\ 3 &0 &0 \end{pmatrix} \)
Answer
For \(a\gt 0\) the signature is \((2{,}1,0)\), for \(a\lt 0\) the signature is \((1{,}2,0)\), and for \(a=0\) the signature is \((1{,}1,1)\).