Abstract The Moore–Penrose inverse celebrated its 100th birthday in 2020, as the notion standing behind term was first defined by Eliakim Hastings Moore 1920 (Bull Am Math Soc 26:394–395, 1920). Its rediscovery Sir Roger Penrose 1955 (Proc Camb Philos 51:406–413, 1955) can be considered a caesura, after which attracted attention it deserves and has henceforth been exploited various research bra...

A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization relates the combinatorial structure of a free matr...

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Draz...