# The quaternion core inverse and its generalizations

## TL;DR

This paper extends various generalized inverses over quaternions, providing their determinantal representations and illustrating the results with a numerical example, thus broadening the understanding of quaternion matrix inverses.

## Contribution

It introduces quaternion versions of the core inverse and related inverses, along with their determinantal formulas, generalizing previous complex matrix results.

## Key findings

- Determinantal representations for quaternion generalized inverses
- Extension of core and related inverses to quaternion matrices
- Numerical example illustrating the formulas

## Abstract

In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field ${\mathbb{H}}$ and get their determinantal representations within the framework of the theory of column-row determinants previously introduced by the author. Since the Moore-Penrose inverse and the Drazin inverse are necessary tools to represent these generalized inverses, we use their determinantal representations previously obtained by using row-column determinants. As the special case, we give their determinantal representations for matrices with complex entries as well. A numerical example to illustrate the main result is given.