Integral powers of order three Latin square matrices

摘要

An order-n Latin square contains numbers, each of which is one of a setof n real numbers, {a_k}_k~n=1,arranged in the form of ann x nmatrix, sucha way that each row and each column of the matrix contains all n numbers.Euler (1707-1783) was the first to study the properties of Latin squares andthey have been the focus of continued attention since. Studies of Latinsquares naturally lead one to elements of group theory and of matrix theory.As will be shown in this note, both of these features may offer interestinginvestigative opportunities for classroom discussions of the permutationgroup on three symbols and of the algebra of the associated permutationmatrices.