% % 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}