  In this paper we study the lower triangular matrix $\mathbb{K}$-algebra
$\Lambda:=\left[\begin{smallmatrix} T & 0 \\ M & U \end{smallmatrix}\right],$
where $U$ and $T$ are basic $\mathbb{K}$-algebras with enough idempotents and
$M$ is an $U$-$T$-bimodule where $\mathbb{K}$ acts centrally. Moreover, we
characterise in terms of $U,$ $T$ and $M$ when, on one hand, the lower
triangular matrix $\mathbb{K}$-algebra $\Lambda$ is standardly stratified in
the sense of the paper "A generalization theory of standardly stratified
algebras I: Standardly stratified ringois"; and on another hand, when $\Lambda$
is locally bounded in the sense of the paper "Locally finite generated modules
over rings with enough idempotents". Finally, it is also studied several
properties relating the projective dimensions in the categories of finitely
generated modules $\mathrm{mod}(U)$, $\mathrm{mod}(T)$ and
$\mathrm{mod}(\Lambda).$
