Let $q$ a nonzero quadratic form on $\mathcal M_2(\Bbb R)$ verifying the relation
$$\forall A,B\in\mathcal M_2(\Bbb R),\; q(AB)=q(A)q(B)$$ The question is to prove that $q=\det$.
What I have tried so far: I proved that $q(I_2)=1$ and if $A$ and $B$ are similar then $q(A)=q(B)$. I proved also that if $A$ is nilpotent then $q(A)=0$ but I can't proceed. Any help?