UPM Institutional Repository

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


Mandangan, Arif and Kamarulhaili, Hailiza and Asbullah, Muhammad Asyraf (2021) Square integer matrix with a single non-integer entry in its inverse. Mathematics, 9 (18). art. no. 2226. 01-11. ISSN 2227-7390


Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, 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∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, 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.

Download File

[img] Text
Square integer matrix with a single non-integer entry in its inverse.pdf

Download (91kB)
Official URL or Download Paper: https://www.mdpi.com/2227-7390/9/18/2226

Additional Metadata

Item Type: Article
Divisions: Institute for Mathematical Research
DOI Number: https://doi.org/10.3390/math9182226
Publisher: MDPI AG
Keywords: Square integer matrix; Inversion of integer matrix; Unimodular matrix; Algebraic number theory; Lattice-based cryptography
Depositing User: Ms. Che Wa Zakaria
Date Deposited: 06 Jan 2023 08:26
Last Modified: 06 Jan 2023 08:26
Altmetrics: http://www.altmetric.com/details.php?domain=psasir.upm.edu.my&doi=10.3390/math9182226
URI: http://psasir.upm.edu.my/id/eprint/95139
Statistic Details: View Download Statistic

Actions (login required)

View Item View Item