Version 1
: Received: 18 February 2024 / Approved: 20 February 2024 / Online: 21 February 2024 (03:45:11 CET)
How to cite:
Jean Paul, P.; Wahid, S. Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications. Preprints2024, 2024021134. https://doi.org/10.20944/preprints202402.1134.v1
Jean Paul, P.; Wahid, S. Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications. Preprints 2024, 2024021134. https://doi.org/10.20944/preprints202402.1134.v1
Jean Paul, P.; Wahid, S. Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications. Preprints2024, 2024021134. https://doi.org/10.20944/preprints202402.1134.v1
APA Style
Jean Paul, P., & Wahid, S. (2024). Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications. Preprints. https://doi.org/10.20944/preprints202402.1134.v1
Chicago/Turabian Style
Jean Paul, P. and Shanaz Wahid. 2024 "Division by Zero: Using Semi-structured Complex Numbers to Find the Inverse of a Singular Matrix and Show Its Applications" Preprints. https://doi.org/10.20944/preprints202402.1134.v1
Abstract
Matrices and their inverses allow accurate calculations to be produced quickly and compactly and can be used to control any process represented by a system of equations. Singular matrices require extensive workarounds (costing time and money) to solve these systems of equations because they have no inverse. Finding their inverse involves division by zero. Nevertheless, recently a new number set called semi-structured complex numbers was developed to enable division by zero in algebraic equations. The aim of this research was to demonstrate that the inverse of a singular matrix can be found using semi-structured complex numbers. This research reveals (1) Singular matrices and their inverses can be used to find a unique solution to a pair of simultaneous equations that appear to have infinitely many solutions or appear to have no solution; (2) Singular matrices map coordinates in projective space to coordinates in Euclidean space and their inverse maps coordinates in Euclidean space to coordinates in projective space; (3) the inverse of a singular matrix proves that parallel lines in Euclidean space do not intersect but in projective space intersect at a “point at infinity” and (4) collinear lines that intersect at infinitely many points in Euclidean space only intersect at one point in projective space.
Keywords
division by zero; inverse of a singular matrix; singular matrix; semi-structured complex numbers; projective geometry
Subject
Computer Science and Mathematics, Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
