# On the Maximal Solution of A Linear System over Tropical Semirings

## TL;DR

This paper introduces methods for solving linear systems over tropical semirings, including reduction techniques, pseudo-inverse computation, and a new Cramer's rule for maximal solutions, with practical Maple implementations.

## Contribution

It presents a novel approach to find the maximal solution of tropical linear systems using a new version of Cramer's rule and reduction techniques.

## Key findings

- Effective reduction of system size through row-column analysis
- A new Cramer's rule for maximal solutions in tropical systems
- Implementation of Maple procedures for pseudo-inverse computation

## Abstract

In this paper, we present methods for solving a system of linear equations, $ AX=b $, over tropical semirings. To this end, if possible, we first reduce the order of the system through some row-column analysis, and obtain a new system with fewer equations and variables. We then use the pseudo-inverse of the system matrix to solve the system if solutions exist. Moreover, we propose a new version of Cramer's rule to determine the maximal solution of the system. Maple procedures for computing the pseudo-inverse are included as well.