Verify that $A\mapsto\det(A)$ is a quadratic form on $\Ma(2,\R)$. Verify that the associated symmetric bi-linear map $g:\Ma(2,\R)\times\Ma(2,\R)\rar\R$, $g(A,A)=\det(A)$ is an inner product and compute its index.
For $x_1,x_2,x_3,x_4\in\R$ and
$$
A=\left(\begin{array}{cc}
x_1&x_2\\
x_3&x_4
\end{array}\right)
\quad\mbox{we have}\quad
\det(A)=x_1x_4-x_2x_3
$$
and thus we need to find the eigen values and multiplicities of the matrix
$$
\frac12\left(\begin{array}{cccc}
0&0&0&1\\
0&0&-1&0\\
0&-1&0&0\\
1&0&0&0
\end{array}\right)
$$
The maxima commands
A:matrix([0,0,0,1],[0,0,-1,0],[0,-1,0,0],[1,0,0,0]);
eigenvalues(A);
show that the above matrix has eigen values $\pm1/2$ each with multiplicity $2$, i.e. $g$ is an inner product with index $2$.