Find the Rank and Signature of a Billinear Form

Question

Let $V \in M_{2 \times 2} ^C$ be the set of all the herimitian of order 2. V is a linear space over $ \Bbb R$.

Check that $q(A) =2det(A) $ is a Square Billinear Form.

In addition, Find the signature and the Rank of $ q(A) $.

I really don't know how to approach this question. I am supposed to find a basis for a general Billinear form, but how do I use the information I'm given?

Thank you,

Alan

Answer 1

Hint: Let $adj(A)$ be the adjugate matrix of $A$. Notice that $adj:M_2\rightarrow M_2$ is a linear transformation. Consider the bilinear form $p(A,B)=tr(Aadj(B))$.