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E. Sk\"oldberg's Morse Theory from an Algebraic Viewpoint and M.J\"ollenbeck's Algebraic Discrete Morse Theory and Applications to CommutativeAlgebra, which is the algebraic generalization of R. Forman's discrete MorseTheory for Cell Complexes, is discussed in the context of general chaincomplexes of free modules.  Using this, we compute the Chevalley-Eilenberg (co)homology of the Liealgebra of all triangular matrices $\frak{sol}_n$ over $\mathbb{Q}$ or$\mathbb{Z}_p$ for large enough prime $p$. We determine the column and row inthe table of $H_k(\frak{sol}_n;\mathbb{Z})$ where the $p$-torsion firstappears. Every $\mathbb{Z}_{p^k}$ appears as a direct summand of some$H_k(\frak{sol}_n;\mathbb{Z})$. Module $H_k(\frak{sol}_n;\mathbb{Z}_p)$ isexpressed by the homology of a chain subcomplex for the Lie algebra of allstrictly triangular matrices $\frak{nil}_n$, using the K\"unneth formula. Allconclusions are accompanied by computer experiments.  Then we generalize some results to the more general Lie algebras of(strictly) triangular matrices $\frak{gl}_n^\prec$ and $\frak{gl}_n^\preceq$with respect to any partial ordering $\preceq$ on $[n]$. Furthermore, thematchings used can be analogously defined for other Lie algebra families andare useful for theoretical as well as computational purposes.

(Co)Homology of Lie Algebras via Algebraic Morse Theory