Algebraic proof methods for identities of matrices and operators: Improvements of Hartwig’s triple reverse order law

•We illustrate a recent method for proving identities of matrices and linear operators.•Computations are done in an abstract ring ignoring domains and codomains.•The theory ensures that identities are also valid in terms of matrices or operators.•We prove and generalize Hartwig’s result on three Moore-Penrose inverses.•Both hand-made and computer-assisted proofs with the package OperatorGB are discussed. When improving results about generalized inverses, the aim often is to do this in the most general setting possible by eliminating superfluous assumptions and by simplifying some of the conditions in statements. In this paper, we use Hartwig’s well-known triple reverse order law as an example for showing how this can be done using a recent framework for algebraic proofs and the software package OperatorGB. Our improvements of Hartwig’s result are proven in rings with involution and we discuss computer-assisted proofs that show these results in other settings based on the framework and a single computation with noncommutative polynomials.