Symmetric latin square of order 9 & 10 ? (focusing the diagonal)

Question

The smallest possible symmetric latin square is the order of 4 which is $$\matrix{1&2&3&4\cr 2&1&4&3\cr 3&4&1&2\cr 4&3&2&1}$$ but i'm also wondering if any order of odd/even latin square can be symmetrical like for example the order of 9 & 10?

Answer 1

For any $n$, $$\matrix{1&2&3&\cdots&n\cr 2&3&4&\cdots&1\cr 3&4&5&\cdots&2\cr \vdots&\vdots&\vdots&\ddots&\vdots\cr n&1&2&\cdots&n-1\cr}$$ is a symmetric Latin square.