Condition number question Please, help me with this problem:
Let $A$ a matrix of orden $100$,
$$A\ =\ \left(\begin{array}{ccccc}
1 & 2 &        &        & \\
  & 1 & 2      &        & \\
  &   & \ddots & \ddots & \\
  &   &        & 1      & 2\\
  &   &        &        & 1
\end{array}\right).$$
Show that $\mbox{cond}_2(A) \geq 2^{99}$.
Thanks in advance.

We consider a general order $n$. Calculate $\|Ax\|/\|x\|$ with $x=(1,1,\ldots,1)^T$ to get a lower bound $p=\sqrt{\frac{9(n-1)+1}{n}}$ for $\sigma_1(A)$. Compute $\|Ax\|/\|x\|$ for $x=\left((-2)^{n-1},(-2)^{n-2},\,\ldots,\,-2,1\right)^T$ to get an upper bound $q=\sqrt{\frac{3}{4^n-1}}$ for $\sigma_n(A)$. Now $\frac pq$ is a lower bound for $\operatorname{cond}_2(A)$ and you may show that it is $\ge2^{n-1}$.
