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%
% This file is part of the gauss package by Manuel Kauers.
% Its purpose is to demonstrate the usage of the package.
% See the gauss-doc.dvi file for further information.
%
\documentclass{article}
\usepackage{amsmath}
\usepackage{gauss}
\usepackage{amssymb} % to import \leadsto
\begin{document}
\section{An Example for Gaussian Elimination}
Consider the real matrix
\[
A(t):=\begin{pmatrix}
1 & 1 & 1 \\
t & 2t & 2 \\
t+1 & 0 & 2t
\end{pmatrix},
\]
which depends on a real valued parameter~$t$.
We want to find all solutions of the homogenous linear system defined by $A$
depending on the parameter~$t$.
Therefore, we use the Algorithm Gauss:
\begin{alignat*}1
A(t)=\begin{gmatrix}[p]
1 & 1 & 1 \\
t & 2t & 2 \\
t+1 & 0 & 2t
\rowops
\add[-t]{0}{1}
\add[-(t+1)]{0}{2}
\end{gmatrix}\kern-6.23pt\leadsto&
\begin{gmatrix}[p]
1 & 1 & 1 \\
0 & t & 2-t \\
0 & -t-1 & t-1
\rowops
\add12
\mult{2}{\cdot(-1)}
\end{gmatrix}\\
\leadsto&\begin{gmatrix}[p]
1 & 1 & 1 \\
0 & t & 2-t \\
0 & 1 & -1
\rowops
\swap12
\add[-t]12
\mult2{:2}
\end{gmatrix}\\
\leadsto&\begin{pmatrix}
1 & 1 & 1 \\
0 & 1 & -1 \\
0 & 0 & 1
\end{pmatrix}.
\end{alignat*}
\subsection*{Conclusion}
We have learned that the matrix $A$ defined above is regular for all real
valued~$t$, and we hopefully also have learned how to use the gauss package.
\end{document}
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