Recently K. Mahler [1] introduced a set of φ(2n) matrices of n rows and columns which form under multiplication the abelian group of the residue classes prime to 2n modulo 2n. These remarkable matrices whose elements 0, 1 and −1, have latent roots and determinants which can be given explicitly. Thus we have new examples of matrices with given elements whose powers, roots, inverses and determinants can be written down precisely. Such matrices are often useful in testing the efficacy of methods for finding these functions for a general matrix.