Let $n$ be a positive integer, and let $V$ be the space of all $n \times n$ matrices over the field of complex numbers. Define a bilinear form $f$ on $V$ by $$f(A, B) = n \, \mbox{tr} (AB) - \mbox{tr}(A) \, \mbox{tr}(B)$$ for all $A$, $B$. Let $V_2$ be the subspace of $V$ consisting of all matrices $A$ such that $\mbox{tr} (A) = 0$ and $ A^* = - A$ (where $A^*$ is the conjugate transpose of $A$). Denote by $f_2$ the restriction of $f$ to $V_2$. Show that $f_2$ is negative definite, i.e., that $$f_2(A,A) < 0$$ for each nonzero $A$ in $V_2$.
My approach:
$f_2(A,A) = n. tr(A^2)$ as $tr(A) =0$. Now $tr(A^2) = tr(-AA^*) $ as $A^* = - A$. Hence $tr(A^2) = - tr(AA^*) = - \sum_{i, j =1} ^{n} A_{ij} A_{ji} ^* = - \sum_{i, j=1}^{n} A_{ij} \bar { A_{ji}} ^T =-\sum_{i, j=1}^{n} A_{ij} \bar{A_{ij}} =- \sum_{i, j=1}^{n} \mid A_{ij} \mid^2$
Am I right? If I am wrong please anybody explain me the right answer. Thanks in advance.