# Reverse-order law for core inverse of tensors

## TL;DR

This paper investigates the reverse-order law for the core inverse of tensors with Einstein product, establishing conditions, exploring mixed-type laws, and applying these results to solve multilinear systems like the Poisson problem.

## Contribution

It introduces new conditions for the reverse-order law of tensor core inverses and applies these to solve multilinear tensor systems, including the Poisson problem.

## Key findings

- Established necessary and sufficient conditions for reverse-order law.
- Presented new results on mixed-type reverse-order law.
- Demonstrated application in solving multilinear systems.

## Abstract

The notion of the core inverse of tensors with the Einstein product was introduced, very recently. This paper we establish some sufficient and necessary conditions for reverse-order law of this inverse. Further, we present new results related to the mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems of tensors via the Einstein product. The prowess of the inverse is demonstrated for solving the Poisson problem in the multilinear system framework.