Do the subordinate matrix norms satisfy $\|AB\|=\|BA\|$? Explain.
I know that $\|A\|_1=\max_{1\leq j\leq n}\sum_{i=1}^{n}|a_{ij}|$ and $\|A\|_2=\max_{1\leq i\leq n}\sum_{j=1}^{n}|a_{ij}|$, but I already tried with these subordinate matrix rules and I do not reach equality. I'm thinking badly? Or is this result true for a different standard than the one I mentioned? Thank you very much.
No. For example, $$\begin{align*} A &= \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}, \\ B &= \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \end{align*}$$ and any matrix norm (subordinate or not) on $\mathbb{R}^2$ $\left(\text{or }\mathbb{C}^2\right)$.