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

Question

let A be a $m \times n$ matrix

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

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?

Answer 1

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 $\pmatrix{0 \\ 1}$).