So würde ich es so aussehen lassen:
![alt text][1]
da die Determinate die selbe bleibt und allein eine Hilfsannotation ergänzt wird.
\documentclass{scrartcl}
%\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{calc,matrix}
\begin{document}
\[
\det(A)
%
=\begin{matrix}\begin{tikzpicture}
\matrix [
matrix of math nodes,
column sep=1em,
row sep=1em,
] (sarrus) {
6 & 0 & -7 \\ 1 & 3 & 8 \\ -2 & 0 & 5 \\
};
% Striche
\path ($(sarrus-1-1.north west)-(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-1.north west)-(0.5em,0)$);
\path ($(sarrus-1-3.north east)+(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-3.north east)+(0.5em,0)$);
\end{tikzpicture}\end{matrix}
%
=\begin{matrix}\begin{tikzpicture}
\matrix [
matrix of math nodes,
column sep=1em,
row sep=1em,
%left delimiter={|} %,right delimiter={|},
] (sarrus) {
6 & 0 & -7 & 6 & 0 \\ 1 & 3 & 8 & 1 & 3\\ -2 & 0 & 5 & -2 & 0\\
};
% Striche
\path ($(sarrus-1-1.north west)-(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-1.north west)-(0.5em,0)$);
\path ($(sarrus-1-3.north east)+(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-3.north east)+(0.5em,0)$);
\path[->,blue] (sarrus-1-1) edge (sarrus-2-2)
(sarrus-2-2) edge (sarrus-3-3)
(sarrus-1-2) edge (sarrus-2-3)
(sarrus-2-3) edge (sarrus-3-4)
(sarrus-1-3) edge (sarrus-2-4)
(sarrus-2-4) edge (sarrus-3-5);
\path[->,red](sarrus-3-1) edge[dashed] (sarrus-2-2)
(sarrus-2-2) edge[dashed] (sarrus-1-3)
(sarrus-3-2) edge[dashed] (sarrus-2-3)
(sarrus-2-3) edge[dashed] (sarrus-1-4)
(sarrus-3-3) edge[dashed] (sarrus-2-4)
(sarrus-2-4) edge[dashed] (sarrus-1-5);
\foreach \c in{3,4,5} \node[blue] at (sarrus-3-\c.south east) {$+$};
\foreach \c in{3,4,5} \node[red] at (sarrus-1-\c.north east) {$-$};
\end{tikzpicture}\end{matrix}
\]
\end{document}
___
Höchstens mit der Notation könnte man sich anfreunden:
![alt text][2]
\documentclass{scrartcl}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{calc,matrix}
\begin{document}
\[
\det(A)
%
=\begin{matrix}\begin{tikzpicture}
\matrix [
matrix of math nodes,
column sep=1em,
row sep=1em,
] (sarrus) {
6 & 0 & -7 \\ 1 & 3 & 8 \\ -2 & 0 & 5 \\
};
% Striche
\path ($(sarrus-1-1.north west)-(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-1.north west)-(0.5em,0)$);
\path ($(sarrus-1-3.north east)+(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-3.north east)+(0.5em,0)$);
\end{tikzpicture}\end{matrix}
%
=\begin{matrix}\begin{tikzpicture}
\matrix [
matrix of math nodes,
column sep=1em,
row sep=1em,
%left delimiter={|} %,right delimiter={|},
] (sarrus) {
6 & 0 & -7 & 6 & 0 \\ 1 & 3 & 8 & 1 & 3\\ -2 & 0 & 5 & -2 & 0\\
};
% Striche
\path ($(sarrus-1-1.north west)-(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-1.north west)-(0.5em,0)$);
\path ($(sarrus-1-3.north east)+(0.5em,0)$) edge[] ($(sarrus-3-3.south east -| sarrus-1-3.north east)+(0.5em,0)$);
\path[->,blue] (sarrus-1-1) edge (sarrus-2-2)
(sarrus-2-2) edge (sarrus-3-3)
(sarrus-1-2) edge (sarrus-2-3)
(sarrus-2-3) edge (sarrus-3-4)
(sarrus-1-3) edge (sarrus-2-4)
(sarrus-2-4) edge (sarrus-3-5);
\path[->,red](sarrus-3-1) edge[dashed] (sarrus-2-2)
(sarrus-2-2) edge[dashed] (sarrus-1-3)
(sarrus-3-2) edge[dashed] (sarrus-2-3)
(sarrus-2-3) edge[dashed] (sarrus-1-4)
(sarrus-3-3) edge[dashed] (sarrus-2-4)
(sarrus-2-4) edge[dashed] (sarrus-1-5);
\foreach \c in{3,4,5} \node[blue] at (sarrus-3-\c.south east) {$+$};
\foreach \c in{3,4,5} \node[red] at (sarrus-1-\c.north east) {$-$};
% Strich rechts
\path ($(sarrus-1-5.north east)+(0.5em,0)$) edge[densely dotted] ($(sarrus-3-5.south east -| sarrus-1-5.north east)+(0.5em,0)$);
\end{tikzpicture}\end{matrix}
~~~= 48
\]
\end{document}
[1]: http://texwelt.de/wissen/upfiles/55555555_50.png
[2]: http://texwelt.de/wissen/upfiles/55555555_51.png
