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 is an element of Z(nxn), 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 is an element of Z(nxn). With the property that det(U) = +/- 1, then U-1 is an element of Z(nxn) is guaranteed such that UU-1 = I, where I is an element of Z(nxn) is an identity matrix. In this paper, we propose a new integer matrix (G) over tilde is an element of Z(nxn), which is referred to as an almost-unimodular matrix. With det((G) over tilde) not equal +/- 1, the inverse of this matrix, (G) over tilde (-1) is an element of R-nxn, 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.