### Abstract

In an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r × r minors of the matrix is one. In the same paper a procedure for constructing a generalized inverse from such a linear combination was also given. In the present paper we show that any reflexive generalized inverse can be obtained by that procedure. We also show that if the integral domain is a principal ideal domain, every generalized inverse can be obtained by that procedure. It is also shown that a matrix A of rank r over an integral domain has Moore-Penrose inverse if and only if the sum of squares of all r × r minors of A is an invertible element of the integral domain.

Original language | English |
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Pages (from-to) | 181-196 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 140 |

Issue number | C |

DOIs | |

Publication status | Published - 15-10-1990 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis

### Cite this

*Linear Algebra and Its Applications*,

*140*(C), 181-196. https://doi.org/10.1016/0024-3795(90)90229-6

}

*Linear Algebra and Its Applications*, vol. 140, no. C, pp. 181-196. https://doi.org/10.1016/0024-3795(90)90229-6

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized inverses over integral domains

AU - Bapat, R. B.

AU - Bhaskara Rao, K. P S

AU - Prasad, K. Manjunatha

PY - 1990/10/15

Y1 - 1990/10/15

N2 - In an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r × r minors of the matrix is one. In the same paper a procedure for constructing a generalized inverse from such a linear combination was also given. In the present paper we show that any reflexive generalized inverse can be obtained by that procedure. We also show that if the integral domain is a principal ideal domain, every generalized inverse can be obtained by that procedure. It is also shown that a matrix A of rank r over an integral domain has Moore-Penrose inverse if and only if the sum of squares of all r × r minors of A is an invertible element of the integral domain.

AB - In an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r × r minors of the matrix is one. In the same paper a procedure for constructing a generalized inverse from such a linear combination was also given. In the present paper we show that any reflexive generalized inverse can be obtained by that procedure. We also show that if the integral domain is a principal ideal domain, every generalized inverse can be obtained by that procedure. It is also shown that a matrix A of rank r over an integral domain has Moore-Penrose inverse if and only if the sum of squares of all r × r minors of A is an invertible element of the integral domain.

UR - http://www.scopus.com/inward/record.url?scp=44949288421&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44949288421&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(90)90229-6

DO - 10.1016/0024-3795(90)90229-6

M3 - Article

VL - 140

SP - 181

EP - 196

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -