
Symmetry proof
R and S are relations on A.
If R and S are asymmetric, then $\displaystyle R \cap S$ are asymmetric.
Can this been done by using generic nxn matrices and showing $\displaystyle M_R$ and $\displaystyle M_S$ where "and" is the the Boolean meet symbol?

I don't know what the boolean meet symbol is but you can try the following:
To prove $\displaystyle R \cap S$ is antisymmetric, assume $\displaystyle (x,y) \in R \cap S$ and $\displaystyle (y,x) \in R \cap S$ and continue using the fact that both R and S are antisymmetric.

"Asymmetric" is not the same as "antisymmetric". I believe, the former means $\displaystyle \forall x,y.\,(x,y)\in R\Rightarrow (y,x)\notin R$. However, the same proof idea works. In fact, if at least one of R and S is asymmetric, then $\displaystyle R\cap S$ is asymmetric.

Oh, I was sure I read antisymmetric; my bad then.
I'm not sure about the asymmetric part, though  I've seen some books define an asymmetric relation as one that is simply not symmetric, so OP should clarify (the same idea will work for all cases)