Normal and positive implies symmetric?

Question

again a matrix question: Suppose I have a matrix that is normal (i.e. $A^TA=AA^T$, thus normal over the real numbers) such that all entries are non negative. Does this imply that the matrix is symmetric? If not, does it yield any other nice property? The symmmetric statement seems to be true for 2x2 matrices, but the 3x3 case is somehow tiresome to compute :(

Thanks a lot

Answer 1

$$A=\pmatrix{2&1&3\\3&2&1\\1&3&2}$$ is a counter-example. It is normal with positive elements, but not symmetric.