(i) Define the terms row-rank, column-rank and rank of a matrix, and state a relation between them.

(ii) Fix positive integers $m, n, p$ with $m, n \geqslant p$. Suppose that $A$ is an $m \times p$ matrix and $B$ a $p \times n$ matrix. State and prove the best possible upper bound on the rank of the product $A B$.