Let $n \in \mathbb { N } ^ { * }$. We define the map $N$ from $\mathcal { M } _ { n } ( \mathbb { R } )$ to $\mathbb { R }$ by the relation:
$$N ( A ) = \sup \left\{ \| A x \| _ { \infty } , \| x \| _ { \infty } \leq 1 \right\}$$
Show that $N$ is a norm on $\mathcal { M } _ { n } ( \mathbb { R } )$ and that if $A = \left[ a _ { i , j } \right] _ { 1 \leq i , j \leq n }$, then
$$N ( A ) = \max _ { i \in \{ 1 , \ldots , n \} } \sum _ { j = 1 } ^ { n } \left| a _ { i , j } \right|$$