Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm doing that wrong.
Next I am asked to find the rank and signature of this quadratic form. This doesn't seem to make any sense as I can't see how to decompose it down into $P^T A P$. I've tried writing out the trace a summation but that doesn't really make sense to me.