The classical adjoint

This paper summarizes the historical background of the notion of the classical adjoint as outlined by Muir, and provides applications of the adjoint to various studies of generalized invertibility of matrices over commutative rings. Specifically, in this setting, the classical adjoint is used to provide a novel proof of von Neumann’s 1936 observation that every matrix over a regular ring is regular, and to provide a necessary and sufficient condition for the existence of the Moore–Penrose inverse of a given matrix. In particular, a representation of the Moore–Penrose inverse is given that leads to an immediate proof of Moore’s 1920 formula specifying the entries of his “reciprocal” in terms of determinants.