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**Generalized inverses over integral domains.**
*(English)*
Zbl 0712.15004

In an earlier paper [Biol. 49, 179–189 (1983; Zbl 0505.15002)] the second author showed that a matrix of rank \(r\) over an integral domain has a generalized inverse if and only if a linear combination of all minors of the matrix is one. He also gave a procedure for constructing a generalized inverse from a linear combination of minors.

This paper shows that any reflective generalized inverse can be obtained by this procedure. In a principal ideal domain the procedure gives all generalized inverses. Finally, a rank \(r\) matrix over an integral domain has a Moore-Penrose inverse if and only if the sum of squares of all \(r\times r\) minors is an invertible element of the integral domain.

Many papers on generalized inverses over various types of rings have been published during the last twenty years. This paper has only four references.

This paper shows that any reflective generalized inverse can be obtained by this procedure. In a principal ideal domain the procedure gives all generalized inverses. Finally, a rank \(r\) matrix over an integral domain has a Moore-Penrose inverse if and only if the sum of squares of all \(r\times r\) minors is an invertible element of the integral domain.

Many papers on generalized inverses over various types of rings have been published during the last twenty years. This paper has only four references.

Reviewer: S.L.Campbell

### MSC:

15A09 | Theory of matrix inversion and generalized inverses |

15B33 | Matrices over special rings (quaternions, finite fields, etc.) |

### Citations:

Zbl 0505.15002
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\textit{R. B. Bapat} et al., Linear Algebra Appl. 140, 181--196 (1990; Zbl 0712.15004)

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### References:

[1] | Bhaskara Rao, K. P.S., On generalized inverses of matrices over principal ideal rings, Linear and Multilinear Algebra, 10, 145-154 (1980) · Zbl 0459.15006 |

[2] | Bhaskara Rao, K. P.S., On generalized inverses of matrices over integral domains, Linear Algebra Appl., 49, 179-189 (1983) · Zbl 0505.15002 |

[3] | K. Manjunatha Prasad and K.P.S. Bhaskara Rao, Integral domains over which every regular matrix has a rank factorization, submitted for publication.; K. Manjunatha Prasad and K.P.S. Bhaskara Rao, Integral domains over which every regular matrix has a rank factorization, submitted for publication. |

[4] | Bruening, J. T., A new formula for the Moore-Penrose inverse, (Uhlig, F.; Grone, R., Current Trends in Matrix Theory (1987), Elsevier Science) · Zbl 0654.15003 |

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