Let $T$ be a linear operator on $\mathbb{R}$ defined by the $2\times2$ matrix $(a_{jk})_{j,k=1}^{2},\: a_{jk}\in\mathbb{R}$. Let $l_{\infty}^{2}$ denote $\mathbb{R}^{2}$ equipped with the norm $||x||_{\infty}=\max(|x_{1}|,|x_{2}|)$ for $x=(x_{1},x_{2})\in\mathbb{R}^{2}$. Let $T:l_{\infty}^{2}\rightarrow l_{\infty}^{2}$. Find the operator norm $||T||$.
I guess it should be $||T||=\max\{|a_{11}|+|a_{12}|,|a_{21}|+|a_{22}|\}$, but I am not sure how to show that. Thank you.