Citation
Mandangan, Arif and Kamarulhaili, Hailiza and Asbullah, Muhammad Asyraf
(2021)
Square integer matrix with a single noninteger entry in its inverse.
Mathematics, 9 (18).
art. no. 2226.
0111.
ISSN 22277390
Abstract
Matrix inversion is one of the most significant operations on a matrix. For any nonsingular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of noninteger entries. These entries could be endless floatingpoint 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 almostunimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single noninteger entry. The almostunimodular matrix could be useful in various areas, such as latticebased cryptography, computer graphics, latticebased 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 almostunimodular matrix could be an alternative to the unimodular matrix.
Download File

Text
Square integer matrix with a single noninteger entry in its inverse.pdf
Download (91kB)


Additional Metadata
Actions (login required)

View Item 