For the matrix norm in Theorem $2$, prove or disprove: $\|AB\|_{\infty}=\|A\|_{\infty}\|B\|_{\infty}$. What about the special case

Question

For the matrix norm in Theorem $2$, prove or disprove: $\|AB\|_{\infty}=\|A\|_{\infty}\|B\|_{\infty}$. What about the special case $\|A^2\|_{\infty}=\|A\|_{\infty}^2$

Taking $A=\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}$ and $B=\begin{bmatrix}2 & 0\\ 0& 0\end{bmatrix}$, we arrive at $2=\|\begin{bmatrix}2 &0 \\ 2 & 0\end{bmatrix}\|_{\infty}=\|AB\|_{\infty}$ and $\|A\|_{\infty}\|B\|_{\infty}=(2)(2)=4$, with which this is not always true, however if $A=B$ as the special case suggests then equality remains.

Is this argument correct? Thank you very much.

Answer 1

You have not proven the case when $A=B$ yet. You have just verify a few examples and the claim could be wrong.

Let $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and see if it holds.