Inspired by the work of Dubut, Goubault, and Goubault-Larrecq (ICALP 2015) on natural homology, Dubut (RAMiCS 2020) introduces finitary diagrams and studies bisimilarity and diagrammatic path logics for them. To this aim, he defines a fragment of the existential theory of the reals, called the existential theory of invertible matrices (ETIM). Using a PSPACE upper bound for this fragment, he proves that for finitary diagrams, bisimilarity can be decided in EXPSPACE and model checking for diagrammatic path logic in PSPACE. We significantly improve both these bounds and settle the complexity of model checking for finitary diagrams. As our first main result, we show that there is an efficient randomized algorithm for ETIM. Combining this with the previous work by Dubut, we obtain an NEXP upper bound for bisimilarity of finitary diagrams and an NP upper bound for diagrammatic path logic. We also provide a matching NP-hardness proof for the latter. The hardness proof introduces constrained layered poset problems, which may be of independent interest, and connects them to finitary diagrams using Gabriel’s theorem for representations of path quivers. For bisimilarity over finite fields, we further improve the upper bound to PSPACE. In ETIM, we quantify over invertible matrices. We finally ask what happens if we instead quantify over matrices from the special linear group, that is, of determinant one. We show that in this case, the resulting fragment is equivalent to the existential theory of the reals, under a mild generalization of the allowed linear constraints.