If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then
$ |AB| \leq m|A||B|$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$
Attempt: $ |AB| = \max\{| \sum_{j=1}^{m}a_{ij}b_{jk} |: i = 1,\ldots,n \text{ and } k = 1,\ldots,p\} \leq \max\{ \sum_{j=1}^{m}|a_{ij}b_{jk}| : i = 1,\ldots,n \text{ and } k = 1,\ldots,p\} \leq m\max\{|a_ij|\}\max\{|b_{jk}|\} = m|A||B| $
Is this correct?