A unimodular matrix is a square integer matrix with determinant . 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 be an unimodular matrix. As the determinant of an integer matrix must also be an integer, is also an integer matrix. It follows that is an integer matrix.
To show the converse, let be an integer square matrix such that is an integer matrix as well. Since and , we can conclude that and are unimodular.