Let $V := M_2(\mathbb{R})$ and let function $f : V \times V \to \mathbb{R}$ be defined by
$$f \left( A, B \right) = 2 \mbox{tr} \left( A B \right) -\mbox{tr} \left( A \right) \mbox{tr} \left( B \right)$$
I showed that it is a bilinear form on $V$ and I found that the rank of its quadratic form $q$ is $3$, and the sign is $(3,0)$ in that way:
$$q\left(A\right)=f\left(A,A\right)=2tr\left(A^2\right)-tr\left(A\right)tr\left(A\right)=2\left(a_{11}^2+2a_{12}a_{21}+a_{22}^2\right)-\left(a_{11}+a_{22}\right)^2$$ The next exercise is to find a subspace of $V$ of maximal dimension that $q|_W$ is nondegenerate.