Let $V$ denote the vector space of $n \times n$ real matrices.

(1) Show that if $\psi(A, B)=\operatorname{tr}\left(A B^{T}\right)$, then $\psi$ is a positive-definite symmetric bilinear form on $V$.

(2) Show that if $q(A)=\operatorname{tr}\left(A^{2}\right)$, then $q$ is a quadratic form on $V$. Find its rank and signature.

[Hint: Consider symmetric and skew-symmetric matrices.]