# Does $\|matrix\| _{op} \geq \|row\|_2 +\|column\|_2$?

let A be a $$m \times n$$ matrix

$$\|A\| _2 := sup_{x \in S^{n-1}, y \in S^{m-1}} $$

let $$r$$ denotes first row of $$A$$ ,and $$c$$ denotes column of $$A$$. then $$\|A\| _2 \geq \|r\|_2 +\|c\|_2$$ where $$\|\|_2$$ denotes Euclidean norm , Is that true?

If $$A$$ is the matrix $$\pmatrix{0 & 0 \\ 0 & 1}$$, then the first row and first column have 2-norm $$0$$, while the operator norm is clearly $$1$$ (attained by $$\pmatrix{0 \\ 1}$$).