Unimodular Matrices and Loops


A unimodular matrix is a square integer matrix with determinant ±1\pm 1. With some basic linear algebra lemmas, we can show that a square integer matrix is unimodular if and only if its inverse exists and the inverse matrix is also an integer matrix.

Let AA be an unimodular matrix. As the determinant of an integer matrix must also be an integer, adj(A)\operatorname{adj}(A) is also an integer matrix. It follows that A1=1detAadj(A)A^{-1} = \frac{1}{\det{A}} \operatorname{adj}(A) is an integer matrix.

To show the converse, let AA be an integer square matrix such that A1A^{-1} is an integer matrix as well. Since det(A)det(A1)=det(AA1)=det(I)=1\det(A) \det(A^{-1}) = \det(A A^{-1}) = \det(I) = 1 and det(A),det(A1)Z\det(A), \det(A^{-1}) \in \mathbb Z, we can conclude that AA and A1A^{-1} are unimodular.