Let $A$ be an $m \times n$ matrix of real numbers. Define the row rank and column rank of $A$ and show that they are equal.

Show that if a matrix $A^{\prime}$ is obtained from $A$ by elementary row and column operations then $\operatorname{rank}\left(A^{\prime}\right)=\operatorname{rank}(A)$.

Let $P, Q$ and $R$ be $n \times n$ matrices. Show that the $2 n \times 2 n$ matrices $\left(\begin{array}{cc}P Q & 0 \\ Q & Q R\end{array}\right)$ and $\left(\begin{array}{cc}0 & P Q R \\ Q & 0\end{array}\right)$ have the same rank.

Hence, or otherwise, prove that

$$\operatorname{rank}(P Q)+\operatorname{rank}(Q R) \leqslant \operatorname{rank}(Q)+\operatorname{rank}(P Q R)$$