This is a question from an example sheet that I think may have a mistake in it.
Show that the set of Hermitian matrices $A \in H_2 (\mathbb{C})$ with Trace$(A\cdot A_{(1,1)})=0$ is a real three dimensional vector space.
So Hermitian $2\times2$ matrices have reals on the diagonal and a complex number and it's conjugate on it's off diagonal. Then the condition Trace$(A\cdot A_{(1,1)})=0$ mean that,
$$(A_{(1,1)})^2 + A_{(2,2)}A_{(1,1)} =0$$
now if $A_{(1,1)} \neq0$ we have matrices of the form;
$$\begin{pmatrix} a & b \\ \bar{b} & -a \end{pmatrix} ~~~~~s.t.~~~~ a \in \mathbb{R}, b\in \mathbb{C}$$
But if $A_{(1,1)}= 0$ then our conditions are also all satisfied and we get matrices of the form;
$$\begin{pmatrix} 0 & b \\ \bar{b} & d \end{pmatrix} ~~~~~s.t.~~~~ d \in \mathbb{R}, b\in \mathbb{C}$$
but summing a matrix of each type we get,
$$ \phi=\begin{pmatrix} a & b \\ \bar{b} & d-a \end{pmatrix} ~~~~~s.t.~~~~ a \in \mathbb{R}, b\in \mathbb{C} $$
But then Trace$(\phi \cdot \phi _{(1,1)}) = da \neq 0$ if $a$ and $d$ aren't. So this can't be a vector space since it isn't closed under addition.
I just wanted to check that I am right about this.
I think I may have misunderstood the notation though, the exact way it is written on the question sheet is $$Trace(AA_{1,1})$$
does this have a usual meaning different to the one I gave?
If I understand your notation correctly, you are right: this isn't a vector space, for the reasons you gave. For example, $A = \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$ and $B = \left(\begin{array}{cc}0&0\\0&1\end{array}\right)$ are both in that set, but $A+B = \left(\begin{array}{cc}1&0\\0&0\end{array}\right)$ is not in that set, as the trace of $1(A+B)$ is $1$, not $0$.