Simple linear algebra justification - clarification

Good day to all.

I have been reading on transposes and came across the following question:

**If A is a n x n matrix where A is non zero can AA^T = 0**

I thought about it and came to the conclusion that the answer is no with the following justification:

Since A is a non zero matrix then at the very least some (A)i,j element is <> 0.

By definition of the transpose this implies that the (A)j,i element is <> 0 and (A)j,i = (A)i,j

Therefore the cross product of the ith row of A with the ith column of A^T will have a minimum value of ((A)i,j)^2

Does this conclusion make sense or have I missed something along the way.

Thanks for your help