TikZ - Explain the computation of the determinant of a square matrix of dimension 3

Question

I would like to show the classical trick to calculate the determinant of a square matrix of dimension 3. Here is my very beginning code.

\documentclass{article}
\usepackage{tikz}
\usepackage{nicematrix}
\begin{document}
$\begin{NiceArray}{|CCC|CC}
    a & b & c & a & b \
    r & s & t & r & s \
    x & y & z & x & y
    %
    \CodeAfter
    %
    \tikz\draw[blue] (1-1.south east) -- (3-3.south east) ;
\end{NiceArray}$
\end{document}

Here is the unwanted output.

I would like the path to be hidden by the text as with the a below, and I need the line to be longer as in the picture below.

Answer 1

The simplest possibility would be to add a node that is filled white on the corresponding entry. Works, but only if the page background is white (which it is not already in your screen shots). So we need to avoid drawing the line in certain stretches. There are several possibilities, out of which I focus on clips and intersections.

You can use an even odd clip. So we first find the boundary path of the entry via fit, and then perform an even odd clip with that path and some large rectangle. The size of the blocked out region depends on the inner sep of the fit node. Everything outside the fit circle and inside the big box will be visible.

\documentclass{article}
\usepackage{tikz}
\usepackage{nicematrix}
\usetikzlibrary{fit}
\tikzset{% https://tex.stackexchange.com/a/76216
even odd clip/.code={\pgfseteorule}}
\makeatletter
\tikzset{% https://tex.stackexchange.com/a/38995

reuse path/.code={\pgfsyssoftpath@setcurrentpath{#1}} 
} 
\makeatother
\begin{document}
$\begin{NiceArray}{|CCC|CC}
    a & b & c & a & b \
    r & s & t & r & s \
    x & y & z & x & y
    %
    \CodeAfter
    \begin{tikzpicture}
     \node[inner sep=0.5pt,circle,fit=(1-1),save path=\pathA]{};
     \clip[even odd clip,reuse path=\pathA] ([xshift=-1cm,yshift=1cm]1-1)
      rectangle ([xshift=1cm,yshift=-1cm]3-3);
     \draw[semithick,blue,-latex,shorten <=-0.8em,shorten >=-0.8em] 
        (1-1.center) -- (3-3.center); 
    \end{tikzpicture}
\end{NiceArray}$
\end{document}

One can thus spare out all the entries, if needed.

\documentclass{article}
\usepackage{tikz}
\usepackage{nicematrix}
\usetikzlibrary{arrows.meta,fit}
\tikzset{% https://tex.stackexchange.com/a/76216
even odd clip/.code={\pgfseteorule}}
\makeatletter
\tikzset{% https://tex.stackexchange.com/a/38995

reuse path/.code={\pgfsyssoftpath@setcurrentpath{#1}} 
} 
\makeatother
\begin{document}
$\begin{NiceArray}{|CCC|CC}
    a & b & c & a & b \
    r & s & t & r & s \
    x & y & z & x & y
    %
    \CodeAfter
    \begin{tikzpicture}
     \node[inner sep=0.5pt,circle,fit=(1-1),save path=\pathA]{};
     \node[inner sep=0.5pt,circle,fit=(2-2),save path=\pathB]{};
     \node[inner sep=0.5pt,circle,fit=(3-3),save path=\pathC]{};
     \clip[even odd clip,reuse path=\pathA] ([xshift=-1cm,yshift=1cm]1-1)
      rectangle ([xshift=1cm,yshift=-1cm]3-3);
     \clip[even odd clip,reuse path=\pathB] ([xshift=-1cm,yshift=1cm]1-1)
      rectangle ([xshift=1cm,yshift=-1cm]3-3);
     \clip[even odd clip,reuse path=\pathC] ([xshift=-1cm,yshift=1cm]1-1)
      rectangle ([xshift=1cm,yshift=-1cm]3-3);
     \draw[semithick,blue,-{Latex[round,length=3.5pt]},shorten <=-0.8em,shorten >=-0.8em] 
        (1-1.center) -- (3-3.center); 
    \end{tikzpicture}
\end{NiceArray}$
\end{document}

Alternatively, intersections can be used.¹

\documentclass{article}
\usepackage{tikz}
\usepackage{nicematrix}
\usetikzlibrary{fit,intersections}
\begin{document}
$\begin{NiceArray}{|CCC|CC}
    a & b & c & a & b \
    r & s & t & r & s \
    x & y & z & x & y
    %
    \CodeAfter
    \begin{tikzpicture}
     \node[inner sep=0.5pt,circle,fit=(1-1),draw,opacity=0,name path=circ1]{};
     \path (1-1.center) -- (3-3.center) 
        coordinate[pos=-0.2] (start) coordinate[pos=1.2] (end);
     \path[name path=line]  (start) to[bend left=0] (end);
     \draw[semithick,blue,-latex,name intersections={of=line and circ1,total=\t}]
       (start) -- \ifnum\t=2 (intersection-1) (intersection-2) --\fi
        (end);
    \end{tikzpicture}
\end{NiceArray}$
\end{document}

If you want to spare out several entries, you can use use loops.

\documentclass{article}
\usepackage{tikz}
\usepackage{nicematrix}
\usetikzlibrary{arrows.meta,fit,intersections}
\begin{document}
$\begin{NiceArray}{|CCC|CC}
    a & b & c & a & b \
    r & s & t & r & s \
    x & y & z & x & y
    %
    \CodeAfter
    \begin{tikzpicture}
     \foreach \X in {1,2,3}{
     \node[inner sep=0.5pt,circle,fit=(\X-\X),draw,opacity=0,
        name path global=circ\X]{};
     }
     \path (1-1.center) -- (3-3.center) 
        coordinate[pos=-0.2] (start) coordinate[pos=1.2] (end);
     \path[name path=line]  (start) to[bend left=0] (end);
     \draw[semithick,blue,-{Latex[round,length=3.5pt]},
        name intersections={of=line and circ1,name=i1,total=\tA},
        name intersections={of=line and circ2,name=i2,total=\tB},
        name intersections={of=line and circ3,name=i3,total=\tC},
        ]
        (start) -- \ifnum\tA=2
        (i1-1) (i1-2) --\fi
        \ifnum\tB=2
        (i2-1) (i2-2) --\fi
        \ifnum\tC=2
        (i3-1) (i3-2) --\fi
         (end);
    \end{tikzpicture}
\end{NiceArray}$
\end{document}

_{¹If you think that I am crazy because I draw a straight line with bend=0: the reason is that otherwise it is harder to sort the intersections: https://tex.stackexchange.com/a/418650.}