Matrix/Column rank/Product/Exercise

Let ${}K$ be a field and let ${}A$ be a ${}m\times n$-matrix and ${}B$ a ${}n\times p$-matrix over ${}K$. Prove the following relationships concerning the rank

$\operatorname {rk} \,AB\leq \operatorname {rk} \,A{\text{ and }}\operatorname {rk} \,AB\leq \operatorname {rk} \,B.$

Prove that equality on the left occurs if ${}B$ is invertible, and equality on the right occurs if ${}A$ is invertible. Give an example of non-invertible matrices ${}A$ and ${}B$ such that equality on the left and on the right occurs.

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