# On Cartesian product of matrices

## TL;DR

This paper explores the Cartesian product of matrices, providing formulas for traces, establishing identities, and applying these concepts to analyze graph properties.

## Contribution

It introduces trace formulas and identities for the Cartesian product of matrices, extending its application to graph theory.

## Key findings

- Derived trace expressions for multiple matrices
- Established identities involving Cartesian products
- Applied Cartesian product to graph property analysis

## Abstract

Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the all one matrix of appropriate order and $\otimes$ is the Kronecker product. In this article, we find the expression for the trace of the Cartesian product of any finite number of square matrices in terms of traces of the individual matrices. Also, we establish some identities involving the Cartesian product of matrices. Finally, we apply the Cartesian product to study some graph-theoretic properties.