Is the following a matrix norm?

Question

Is the following a norm on set of symmetric matrices of order $n$ :

$$|||A|||=\max_{1\le i\le n}\{||r_i|| \}, $$ where $||r_i||$ is usual Euclidean norm of $i$-th row vector $r_i$ ?

I can see that it satisfies the properties

$1.$ $|||A||| \ge 0$ and equality occurs iff $A=0.$

$2.$ $|||cA|||=c|||A|||$ for all complex numbers $c$.

$3.$ $|||A+B|||\le |||A|||+|||B|||$

but I am not sure about $|||AB||| \le |||A|||\text{ }|||B|||.$

Answer 1

It doesn't satisfy the last property. See what happens when $A=B=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$.