a) For any matrix $M \in M _ { n } ( \mathbb { C } )$ and any real number $C > 0$, show the equivalence
$$\| M \| \leqslant C \Longleftrightarrow \forall x \in \mathbb { C } ^ { n } : \| M x \| _ { 1 } \leqslant C \| x \| _ { 1 } .$$
b) Show that the map $M \longmapsto \| M \|$ is a norm on $M _ { n } ( \mathbb { C } )$.