Let $m,n \in\mathbb{N}$ and $ P,Q\in M_{n\times m}(\mathbb{R})$. Then which of the following is/are not possible?

Question

$(a)$ ${rank}(PQ)=n$

$(b)$ $rank (QP)=m$

$(c)$ $rank(PQ)=m$

$(d)$ $rank(QP)=k$ , where $k$ is the smallest integer value of $\frac{m+n}{2}$.

This is not a homework , it is asked in my exam today.

Answer 1

In order for $PQ$ or $QP$ to even make sense, you would have to have $m=n$. Then these are square $m\times m$ (or $n \times n$, same thing) matrices. Also note that then $\frac{m+n}{2}=\frac{m+m}{2}=m$. Is it possible for $PQ$ and $QP$ to have rank $m$?