Let $W = \left\{ A \in M_{3\times 3}(\mathbb{R}) : A^t + A \in \mathbb{R} \cdot I \right\}$ where $I$ is the identity matrix. Consider the quadratic form $$Q(A) = \mbox{trace} (A)^2 - \mbox{trace} \left( A^2 \right).$$ Show that the signature of $Q$ is $(+\ + \ +\ +)$.
My attempt: I have calculated that if $A=(a_{ij})$, then $$Q(A)=2(a_{11}a_{22}+a_{11}a_{33}+a_{22}a_{33}-a_{12}a_{21}-a_{13}a_{31}-a_{23}a_{32}).$$ Also, as $A\in V$, we must have $a_{ij}=-a_{ji}$ for each $i, j$ whenever $i\neq j$. How to proceed further? Any hint is highly appreciated.
Your work so far is helpful. Note also that if $A \in W$, then $a_{11} = a_{22} = a_{33} = k$. With that and the fact that $a_{ij} = - a_{ji}$ for $i \neq j$, we can rewrite your expression as $$ Q(A) = 2(3a_{11}^2 + a_{12}^2 + a_{13}^2 + a_{23}^2). $$ Using this new expression for $Q(A)$, it's easy to show that $Q(A) \geq 0$ for all $A$ and that $Q(A) = 0$ if and only if $A = 0$, which is enough to deduce that the signature of $Q$ is $(+\,+\,+\,+)$.