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The paper investigates the class of square matrices which have idempotent Moore–Penrose inverse. A number of original characteristics of the class are derived and new properties identified. Additionally, some isolated results scattered in the literature are recaptured and rederived in an extended or generalized form. The research was inspired by some matrix equations (occurring, e.g. in physics) which either directly involve matrices with an idempotent Moore–Penrose inverse or whose solvability under the assumption that the matrices involved have an idempotent Moore–Penrose inverse provides an emerging insight into the corresponding theory.

On matrices whose Moore–Penrose inverse is idempotent