Proving matrix norm to be submultiplicative?

Question

Consider the matrix norm $\|\mathbf{A}\|_\chi := n\max|a_{ij}|$.

How would I prove it satsifies the property that $\|\mathbf{AB}\| \leq\|\mathbf{A}\|\cdot\|\mathbf{B}\|$.

Answer 1

There are some $1\leq i,j\leq n$ such that:

$||AB||=n|\sum\limits_{k=1}^n a_{ik}b_{kj}|\leq n\sum\limits_{k=1}^n |a_{ik}|\cdot |b_{kj}|\leq n\sum\limits_{k=1}^n \frac{||A||}{n}\cdot\frac{||B||}{n}=n^2 \frac{||A||}{n}\cdot\frac{||B||}{n}=||A||\cdot ||B||$