Biswas, R.Roy, F.2026-02-032025Linear Algebra and Its Applications, 2025, 705, , pp. 185-206243795https://doi.org/10.1016/j.laa.2024.11.005https://idr.nitk.ac.in/handle/123456789/20466For two idempotent matrix P,Q?Cn×n, let alg(I<inf>n</inf>,P,Q) denote the smallest subalgebra of Cn×n that contains P,Q and the identity matrix I<inf>n</inf>. This paper provides a complete classification of alg(I<inf>n</inf>,P,Q) without imposing any restrictions on P and Q. As a result of this classification, the issue of group invertibility within alg(I<inf>n</inf>,P,Q) is fully resolved. © 2024 Elsevier Inc.Complete classificationFinite-dimensional algebrasGroup inversionIdempotent matrixIdentity matricesInvertibilityMatrix PSubalgebrasTwo projection