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\author{Harald R�ck}

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\textbf{First name:} Harald \\
\textbf{Last name:} R"ock \\
\textbf{Date:} \today \\
\textbf{Homework number:} 2 \\
\textbf{Homework Title:} Exercise 2.33 


\paragraph{Problem description:} ~\\
Prove or give a counterexample: \\
If $\mathbf{A}$ is a nonsingular matrix, then
$||\mathbf{A}^{-1}|| = ||\mathbf{A}||^{-1}$

\paragraph{Problem solution:} ~\\
If $||\mathbf{A}^{-1}|| = ||\mathbf{A}||^{-1}$ then 
$||\mathbf{A}^{-1}||\cdot||\mathbf{A}|| = 1$. From that, each condition
number must be 1. 

The counterexample is Example 2.4 and 2.5.

$\mathbf{A} = \left[
\begin{array}{ccc}
    2 & -1  & 1 \\
    1 & 0   & 1 \\ 
	3 & -1  & 4 
\end{array} \right]$

\paragraph{Results}~\\
(compare with Example 2.4 and 2.5) \\
$||\mathbf{A}||_1 = 6$ ~;~  $||\mathbf{A}^{-1}||_1 = 4.5$
$\Rightarrow 1/6 \neq 4.5$


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