Suppose we have $A,B \in \operatorname{Mat}(n, \mathbb{R})$, and let $\alpha(A,B) = \text{Tr}(AB)$ and $\beta(A,B) = \text{Tr}(AB^T)$. Then is it possible to show that both $\alpha$ and $\beta$ cannot be bilinear forms? If not then what can be shown as a counter-example?
Furthermore, how can I show that only $\beta$ gives an inner product space on $\operatorname{Mat}(n, \mathbb{R})$ but $\alpha$ doesn't? How can it be shown that β(A,A)≥0 but that this is not true for α ? I assume this is the condition for the inner product which α does not satisfy but β does?
My attempts
I have tried to use the definition of bilinear forms paired with proof by contradiction, I first assumed that both α and β are bilinear forms and then have tried to use the definition to show this cannot be the case at the same time, however I am not sure how I can do this in a general form.
I did this for both α and β and found that they both satisfy the conditions for being bilinear forms, but I fail to understand how one of α or β being a bilinear form contradicts the other being bilinear too.