Is every "weakly square" matrix either a $0$ matrix, or a square matrix?

Question

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly,

• every square matrix is weakly square, and
• every zero matrix is weakly square.

Question. Are these the only examples of weakly-square matrices?

Remark. I got the idea from Donald Reynolds answer here.

Answer 1

A non-square matrix $A$ is weakly square if and only if neither $A$ nor $A^T$ has full rank, which is to say iff $\operatorname{rank}(A)<\min\{m,n\}$.

The key to this observation is to note that $$ \operatorname{rank}(A^TA)= \operatorname{rank}(A)= \operatorname{rank}(A^T)= \operatorname{rank}(AA^T) $$