Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

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