# Generalized inverses over integral domains. II. group inverses and Drazin inverses

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### 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 31-47 17 Linear Algebra and Its Applications 146 C https://doi.org/10.1016/0024-3795(91)90018-R Published - 15-02-1991

### Fingerprint

Group Inverse
Drazin Inverse
Integral domain
Generalized Inverse
Polynomials
Necessary Conditions
Sufficient Conditions
Square matrix
Invertible
Continuation
Minor
If and only if
Polynomial
Coefficient

### All Science Journal Classification (ASJC) codes

• Algebra and Number Theory
• Numerical Analysis

### Cite this

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

In: Linear Algebra and Its Applications, Vol. 146, No. C, 15.02.1991, p. 31-47.

Research output: Contribution to journalArticle

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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.

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