  We analyze the forcing notion $\mathcal P$ of finite matrices whose rows
consists of isomorphic countable elementary submodels of a given structure of
the form $H_{\theta}$. We show that forcing with this poset adds a Kurepa tree
$T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only
continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level
set of any antichain in $T$ is not stationary in $\omega_1$.
