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Formulae for the Moore–Penrose inverse M+ of rank-one-modifications of a given m×n complex matrix A to the matrix M=A+bc*, where b and c* are nonzero m×1 and 1×n complex vectors, are revisited. An alternative to the list of such formulae, given by Meyer [SIAM J. Appl. Math. 24 (1973) 315] in forms of subtraction–addition type modifications of A+, is established with the emphasis laid on achieving versions which have universal validity and are in a strict correspondence to characteristics of the relationships between the ranks of M and A. Moreover, possibilities of expressing M+ as multiplication type modifications of A+, with multipliers required to be projectors, are explored. In the particular case, where A is nonsingular and the modification of A to M reduces the rank by 1, such a possibility was pointed out by Trenkler [R.D.H. Heijmans, D.S.G. Pollock, A. Satorra (Eds.), Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker, Kluwer, London, 2000, p. 67]. Some applications of the results obtained to various branches of mathematics are also discussed. [Copyright &y& Elsevier]

It is shown that if P1 and P2 are orthogonal projectors, then a product having P1 and P2 as its factors is equal to another such product if and only if P1 and P2 commute, in which case all products involving P1 and P2 reduce to the orthogonal projector P1P2. This is a generalization of a result by Baksalary and Baksalary [Linear Algebra Appl. 341 (2002) 129], with the proof based on a simple property of powers of Hermitian nonnegative definite matrices. [Copyright &y& Elsevier]

The problem of characterizing situations, in which a linear combination C=c1A+c2B of an idempotent matrix A and a tripotent matrix B is an idempotent matrix, is thoroughly studied. In two particular cases of this problem, when either B or −B is an idempotent matrix, a complete solution follows from the main result in [Linear Algebra Appl. 321 (2000) 3]. In the present paper, a complete solution is established in all the remaining cases, when B is an essentially tripotent matrix in the sense that both idempotent matrices B1 and B2 constituting its unique decomposition B=B1−B2 are nonzero. The problem is considered also under the additional assumption that the differences A−B1 and A−B2 are Hermitian matrices. This obviously covers the case when A, B1, and B2 are Hermitian themselves, when the problem can be interpreted from a statistical point of view. [Copyright &y& Elsevier]