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Let $|| \cdot ||$ be a matrix norm on $m \times n$ matrices, which is not assumed to be submultiplicative. Is it true that $||A|| \le \max|a_{ij}|\cdot ||(1)||$ where $(1)$ denotes the matrix with all entries equal to 1? I tried to use triangle inequality but I don't know how to prove it.

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It is incorrect when $mn>1$. A simple counter-example: for $A=(a_{ij})$, let $$\|A\|=|a_{11}|+\sum_{ij}|a_{ij}-a_{11}|.$$

Remark: When $m=n>1$, even if $\|\cdot\|$ is submultiplicative, the conclusion is still incorrect . To provide a counter-example, you may choose $\|\cdot\|$ as defined in $(4)$ here with $P=\lambda\cdot(1)$, where $\lambda>1$ is a large constant determined by the norm $\|\cdot\|'$ in $(4)$.

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