Max norm can be defined as: $$\ ||A||_{1\infty} = \max_{i,j}|a_{ij}|$$
The hint i was given is that for certain c we have$\ ||Ax|| \le c||x||$ and so the following $\ ||A|| \le c$. To be honest i have no idea how to use this hint to prove above equality. I know that i can use the formula $$\ ||A||_{pq} = \max\frac{||Ax||_q}{||x||_p}$$Of course $\ x \in \mathbb{R}^n $ and $\ x \ne0$. However i don't know how to go further with this. Could you please help me?
I finally got your hint... as written, is very unclear, but what they wanted to says is that : If $c$ is s.t. $\|Ax\|_q\leq c\|x\|_p$ for all $x$, then $\|A\|_{pq}\leq c$.
Notice that $$\|A\|_{pq}=\max_{\|x\|_p=1}\|Ax\|_q=\max_{x\neq 0}\frac{\|Ax\|_q}{\|x\|_p}.$$
So, in your case, $(p,q)=(1,\infty )$, and the statement is equivalent than proving that if $\|Ax\|_\infty \leq \max_{i,j}|a_{ij}|$ for all $\|x\|_1=1$, then $\|A\|_{1\infty }\leq \max_{i,j}|a_{ij}|$. Which will implies by your hint, that $\|A\|_{1,\infty }\leq \max_{i,j}|a_{ij}|$ holds.
Let $x\in \mathbb R^n$ be s.t. $\|x\|_{1}=1$. Then, clearly \begin{align*} \|Ax\|_\infty &=\max_{1\leq i\leq n}\{|a_{i1}x_1+...+a_{in}x_n|\}\\ &=|a_{j1}x_1+...+a_{jn}x_n|,\ \ \ \ \text{for some $j$}\\ &\leq \|x\|_1\max_{1\leq i\leq n}|a_{ji}|\\ &\leq \max_{i,j}|a_{ij}|, \end{align*} as wished.
The fact that $\|A\|_{1,\infty }=\max_{i,j}|a_{ij}|$ whenever $\|x\|_1=1$ is now straightforward and left as a Homework ;-).