In an exercice, I have the following:
$B(M,N)=\frac{1}{2}Tr(M\tilde{N})$ where $B:M_2(\Bbb C)^2 \to \Bbb C$ and $\tilde{N}=^t(com N)$
Prove B is nondegenerate.
I tried to prove that $M\in M_2(\Bbb C)$ is not orthogonal to itself:
$B(M,M)=det\ M$ then $B(M,M)=0 \Rightarrow det\ M=0 \not \Rightarrow M=0$ so there must be another way.
Thank you for your help.