Let each row and each column of a $n \times n$ matrix $A$ be a permutation of $\{1, 2,...n\}$ and let $A$ be symmetric.
(a) If $n$ is odd, prove that each of $1, 2,..., n$ occurs on the principle diagonal of $A$.
(b) For every even number $n$, show that there exists an $A$ in which not all of $1, 2,.... n$ appear on the diagonal.
Hint(a): Consider parity. Each integer must appear an odd number of times. Hence, it must appear on the diagonal.
Hint(b): Construct this yourself, it's not hard.