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

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
$$
\operatorname{rank}(A^TA)=
\operatorname{rank}(A)=
\operatorname{rank}(A^T)=
\operatorname{rank}(AA^T)
$$
