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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}
\ No newline at end of file