Find if a form is symmetric or skew-symmetric

Question

Consider the set of all n × n matrices in R. Given the defined function

Φ: $M$(n,n)× $M$(n,n) → R , which

Φ(A,B) = $tr$(A$^T$JB) ,

where J is a skew-symmetric n × n matrix , define if Φ is a bilinear form, and in case it is define if it is symmetric or skew-symmetric.

So far, I've already proved that Φ is a bilinear form. I know that a form is symmetric if:

Φ(A,B) = Φ(B,A)

and skew-symmetric if

Φ(A,B) = -Φ(B,A)

I tried to prove it by doing:

Φ(A,B) = $tr$(A$^T$JB)

Φ(B,A) = $tr$(B$^T$JA)

but I'm not sure how to proceed in order to prove if Φ is symmetric or skew-symmetric.

Answer 1

Hint: $$ \DeclareMathOperator{\tr}{tr} \tr(A^TJB) = \tr([A^TJB]^T) = \tr(B^TJ^TA) = -\tr(B^TJA) $$