  The main purpose of this note is to present a matrix version of Neeman's
forcing with sequences of models of two types. In some sense, the forcing
defined here can be seen as a generalization of Asper\'o-Mota's pure side
condition forcing from arXiv:1203.1235 and arXiv:1206.6724. In our case, the
forcing will consist of certain symmetric systems of countable and uncountable
elementary submodels of some $H(\kappa)$, which will be closed by the relevant
intersections, in Neeman's sense. The goal of this note is to show that this
forcing preserves all cardinals by showing that it is strongly proper for the
class of countable submodels of $H(\kappa)$ and for the class of countably
closed submodels of $H(\kappa)$ of size $\aleph_1$, and that it has has the
$\aleph_3$-cc. Moreover, we will show that it preserves
$2^{\aleph_1}=\aleph_2$.
