Let $A$ be an $m \times n$ matrix of rank $r$. (a) Suppose $C=(A B)$ is an $m \times k$ matrix, $k>n$, whose first $n$ columns are the same as the columns of $A$. Prove that $\operatorname{rank} C \geq \operatorname{rank} A$. Give an example with $\operatorname{rank} C=\operatorname{rank} A$; with $\operatorname{rank} C>\operatorname{rank} A$. (b) Let $E=\left(\begin{array}{c}A \\ D\end{array}\right)$ be a $j \times n$ matrix, $j>m$, whose first $m$ rows are the same as those of $A$. Prove that $\operatorname{rank} E \geq \operatorname{rank} A$. Give an example with $\operatorname{rank} E=\operatorname{rank} A$; with $\operatorname{rank} E>\operatorname{rank} A$.