Suppose $A=(a_{ij})$ is an $n\times n$ real matrix and define $T(A)=\max\{|a_{ij}|\}$, where the maximum is taken over $1\leq i,j \leq n$.
I know how to show that $T(AB)\leq nT(A)T(B)$ for all $A$ and $B$.
Show that $T(A^{r})\leq n^{r-1}(T(A))^{r}$ for all $A$ and all $r\geq 1, r\in\mathbb{R}$.