Unimodular matrix

In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b are both integer, and M is unimodular, has an integer solution. The unimodular matrices of order n form a group, which is denoted .

Examples of unimodular matrices

Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular:

Further:

Concrete examples include:

Total unimodularity

A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A totally unimodular matrix need not be square itself. From the definition it follows that any totally unimodular matrix has only 0, +1 or −1 entries. The opposite is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular.