Find the condition number of
$$A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{bmatrix}$$
The way they want us to do it is to find $\|A\|$, then $\|A^{-1}\|$ and then $\|A\| \|A^{-1}\|$. However, I can't manage to find $\|A\|$. I've tried using Lagrange multipliers, but I get four different $\lambda_i$ for each $x_i$ when I try to solve $\max\limits_{x \neq 0} \frac{||Ax||}{||x||}$.
Our optimization problem can be framed as minimize $\|Ax\|^2$ subject to $\|x\|^2 = 1$. In other words, we have $$ f(x) = 10^{-4}x_1^2 + 10^{-6} x_2^2 + 10^8x_3^2 + 100\,x_4^2\\ g(x) = x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1 $$ Setting $f' - \lambda g' = 0$ yields $$ (1 - 10^{-4}\lambda)x_1 = 0\\ (1 - 10^{-6}\lambda) x_2 = 0\\ (1 - 10^8\lambda)x_3 = 0\\ (1 - 100 \lambda )x_4 = 0 $$ This should indeed lead to four solutions for $\lambda$, and the corresponding vectors $x = (1,0,0,0); x = (0,1,0,0)$ and so on.
Once we have the $\lambda_i$, we have to check each of the corresponding solutions. The greatest value for $f(x)$ comes from $\lambda = 10^{-8}$, for which we attain $f(x) = 10^8$.
Thus, the maximum is $\|Ax\| = \sqrt{10^8} = 10^4$.