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In this paper, we prove that the lower triangular matrix category $\Lambda=\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix}\right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary$\mathrm{Hom}$-finite Krull-Schmidt $K$-categories and $M$ is a $\mathcalU\otimes_K \mathcal T^{op}$-module that satisfies suitable conditions, isquasi-hereditary in the sense of \cite{LGOS1} and \cite{Martin}. Moreover, wesolve the problem of finding quotients of path categories isomorphic to thelower triangular matrix category $\Lambda$, where $\mathcal T=K\mathcal{R/J}$and $\mathcal U=K\mathcal{Q/I}$ are path categories of infinity quivers moduloadmissible ideals. Finally, we study the case where $\Lambda$ is a pathcategory of a quiver $Q$ with relations and $\mathcal U$ is the full additivesubcategory of $\Lambda$ obtained by deleting a source vertex $*$ in $Q$ and$\mathcal T=\mathrm{add} \{*\}$. We then show that there exists an adjoint pairof functors $(\mathcal R, \mathcal E)$ between the functor categories$\mathrm{mod} \ \Lambda$ and $\mathrm{mod} \ \mathcal U$ that preserveorthogonality and exceptionality; see \cite{Assem1}. We then give some examplesof how to extend classical tilting subcategories of $\mathcal U$-modules toclassical tilting subcategories of $\Lambda$-modules.

Triangular Matrix Categories over path Categories and Quasi-hereditary Categories, as well as one point extensions by Projectives