$V = M_{2}(\mathbb{C})$, $K = \mathbb{C}$ and
$$ S = \left\{ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \in V : a_{ij} = \overline{a_{ji}}\text{ for } i, j = 1, 2 \right\} .$$
I'd like to understand better how to work with complex and matrix to check this question. I know there are 3 rules, but id like how to apply in cases like that. Thanks in advance, ill be glad for explanations.
For $i \in \{1,2\}$, $\overline{a_{ii}} = a_{ii}$ implies that $a_{ii}$ is a real number, so $$S = \left\{ \begin{pmatrix} x & z \\ \overline{z} & y \end{pmatrix} \in V : x,y \in \mathbb R \textrm{ and } z \in \mathbb C \right\}.$$ Now, is the sum of two matrices in $S$ a matrix in $S$? and the scalar multiplication of a matrix in $S$ with a complex number?