Product of a primitive matrix and its transpose.

Question

Is it true that if $A$ is a nonnegative primitive matrix, then $AA^T$ is also primitive? Obviously $A^T$ is primitive but in general product of primitive matrices is not primitive. Any hint?

Answer 1

No. Counterexample: $$ A=\pmatrix{0&1&1&0\\ 0&1&1&0\\ 1&0&0&1\\ 1&0&0&1}, \ A^2=\pmatrix{1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1}, \ AA^T=\pmatrix{2&2&0&0\\ 2&2&0&0\\ 0&0&2&2\\ 0&0&2&2}. $$