Matrix one-norm and infinity-norm

Question

Help me please to find $3\times 3$ matrices $A$ and $B$ under following conditions: $\left \| A \right \|_{\infty }=4\left \| A \right \|_{1}, A \neq 0$ $\left \| B \right \|_{1}=4\left \| B \right \|_{\infty }, B \neq 0$.

Thanks a lot!

Answer 1

This is not possible, because

Claim: For a $A$ being an $n\times n$ matrix $A$ we have $$1/n\cdot \|A\|_1\le \|A\|_\infty \le n\cdot \|A\|_1$$

Proof: $$ \begin{align} \|A\|_\infty &= \max_i \sum_j |a_{ij}| \le \sum_j\sum_{i} |a_{ij}|\le \sum_j \max_{k} \sum_i|a_{ik}|=n\|A\|_1 \end{align} $$ This shows the first inequality. The second follows by interchanging the roles of rows and columns in this calculation. $\square$