### Abstract

This is a continuation of an earlier paper by the authors on generalized inverses over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient conditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain R has a group inverse if and only if the sum of all r × r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from R.

Original language | English |
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Pages (from-to) | 31-47 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 146 |

Issue number | C |

DOIs | |

Publication status | Published - 15-02-1991 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis

### Cite this

*Linear Algebra and Its Applications*,

*146*(C), 31-47. https://doi.org/10.1016/0024-3795(91)90018-R

}

*Linear Algebra and Its Applications*, vol. 146, no. C, pp. 31-47. https://doi.org/10.1016/0024-3795(91)90018-R

**Generalized inverses over integral domains. II. group inverses and Drazin inverses.** / Manjunatha Prasad, K.; Bhaskara Rao, K. P S; Bapat, R. B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized inverses over integral domains. II. group inverses and Drazin inverses

AU - Manjunatha Prasad, K.

AU - Bhaskara Rao, K. P S

AU - Bapat, R. B.

PY - 1991/2/15

Y1 - 1991/2/15

N2 - This is a continuation of an earlier paper by the authors on generalized inverses over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient conditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain R has a group inverse if and only if the sum of all r × r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from R.

AB - This is a continuation of an earlier paper by the authors on generalized inverses over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient conditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain R has a group inverse if and only if the sum of all r × r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from R.

UR - http://www.scopus.com/inward/record.url?scp=0039429209&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039429209&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(91)90018-R

DO - 10.1016/0024-3795(91)90018-R

M3 - Article

AN - SCOPUS:0039429209

VL - 146

SP - 31

EP - 47

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -