Permutations Everywhere
If A = {aij } is a symmetric (i.e., aij = aji) n by n matrix with n odd, and each row of the matrix is a permutation of the integers 1, 2, 3, … , n, prove that the main diagonal of A is also a permutation of 1, 2, 3, … , n.
Permutations Everywhere
If A = {aij } is a symmetric (i.e., aij = aji) n by n matrix with n odd, and each row of the matrix is a permutation of the integers 1, 2, 3, … , n, prove that the main diagonal of A is also a permutation of 1, 2, 3, … , n.
Each number from 1 to n appears exactly n times. Because of the symmetry, it appears an even number of times away from the diagonal. Since n is odd, it should appear on the diagonal as well. Thus, all numbers appear on the diagonal. Since there are n numbers on the diagonal, each number from 1 to n appears on the diagonal exactly once.