\section*{Homework 4 \qquad\qquad\qquad\rm \normalsize Due Wed May 7}

\begin{enumerate}

\item With the help of \Matlab, complete this table in terms of $|\mu|$:
      \[
         \cond\pmat{1 & \\ \mu & 1} \approx
           \left\{
             \begin{array}{ll}
                \dots,  & |\mu| \ll  1,
             \\ 2.6180, & |\mu|  = \,1,
             \\ \dots,  & |\mu| \gg  1.
             \end{array}
           \right.
      \]

\item If $M_k = \pmat{1 & \\ \mu_k & 1}$, what is $M_1 M_2$?

\item If $M_1 = \pmat{1 & \\ 50 & 1}$, $M_2 = \pmat{1 & \\ 50 & 1}$,
      what are $M_1 M_2$ and $\cond(M_1 M_2)$?

\item How does the previous $\cond(M_1 M_2)$ compare with the bound
      \[
        \cond(M_1 M_2) \le \cond(M_1)\cond(M_2)?
      \]

\item If
      \[
        L_1 = \pmat{1 \\ &1 \\ & &1 \\ 10&  &  &1}, \quad
        L_2 = \pmat{1 \\ &1 \\ & &1 \\   &10&  &1}, \quad
        L_3 = \pmat{1 \\ &1 \\ & &1 \\   &  &10&1},
      \]
      what are $L_1 L_2 L_3$ and  $\cond(L_1 L_2 L_3)$?

\item How does the previous $\cond(L_1 L_2 L_3)$ compare with the bound
      \[
        \cond(L_1 L_2 L_3) \le \cond(L_1)\cond(L_2)\cond(L_3)?
      \]

\end{enumerate}