Square integer matrix with a single non-integer entry in its inverse

• Main Authors: Arif Mandangan, Hailiza Kamarulhaili, Muhammad Asyraf Asbullah
• Format: Article
• Language: English; English
• Published: Molecular Diversity Preservation International (MDPI) 2021
• Subjects: QA440-699 Geometry. Trigonometry. Topology; T57-57.97 Applied mathematics. Quantitative methods
• Online Access: https://eprints.ums.edu.my/id/eprint/36114/1/ABSTRACT.pdf; https://eprints.ums.edu.my/id/eprint/36114/2/FULL%20TEXT.pdf; https://eprints.ums.edu.my/id/eprint/36114/; https://doi.org/10.3390/math9182226
• Institution: Universiti Malaysia Sabah

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A?Z^{nÃn}, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U?Z^{nÃn}. With the property that det(U)=±1, then U^{-1}?Z^{nÃn} is guaranteed such that UU^{-1}=I, where I?Z^{nÃn} is an identity matrix. In this paper, we propose a new integer matrix \tilde{G}?Z^{nÃn}, which is referred to as an almost-unimodular matrix. With det(\tilde{G})?±1, the inverse of this matrix, \tilde{G}^{-1}?R^{nÃn}, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.