Example 11.11

Let $A=[ −1−101−111011112010]$ . This example illustrates the process just described that derives GEPP.

Start: $L=[ 1000010000100001],P=[ 1000010000100001]$ i = 1: Exchange rows 1 and 4. Eliminate entries at indices (2, 1) − (4, 1):

$\begin{array}{l}{E}_{1}={E}_{41}\left(-\frac{1}{2}\right){E}_{31}\left(\frac{1}{2}\right){E}_{21}\left(-\frac{1}{2}\right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0.5& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ -0.5& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0.5& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0.5& 1& 0& 0\\ -0.5& 0& 1& 0\\ 0.5& 0& 0& 1\end{array}\right]\\ {P}_{1}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\end{array}\right]\\ {E}_{1}{P}_{1}A=\left[\begin{array}{cccc}2& 0& 1& 0\\ 0& 1& 1.5& 0\\ 0& 1& 0.5& 1\\ 0& -1& 0.5& 1\end{array}\right]\end{array}$ i = 2: A row exchange is not ...

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