I am trying to prove that $$\mbox{cond}_2(A)\ =\ \inf_{E\in\mathscr{E}}\mbox{cond}_2(E),\;\;\; \mbox{ where }\;\; A = \left(\begin{array}{cc} 100 & 99\\99 & 98 \end{array}\right).$$ And $\mathscr{E}$ is the set of matrices of order 2, whose elements $a_{ij}$ are integers satisfying $0\leq a_{ij}\leq 100$.
In order to prove it, it's clear that $\mbox{cond}_2(A) \geq \inf\limits_{E\in\mathscr{E}}\mbox{cond}_2(E)$, and I have showed that for every general $A$ of order 2, that $$\mbox{cond}_2(A)\ =\ \sigma + \sqrt{\sigma^2-1},\;\;\; \mbox{ with }\;\; \sigma = \frac{\sum\limits_{i,j=1}^2|a_{ij}|^2}{2|\det(A)|},$$ but I don't know how to prove that $\mbox{cond}_2(A) \leq \inf\limits_{E\in\mathscr{E}}\mbox{cond}_2(E)$ for that particular $A$. Please, somebody help me. Thanks in advance.