How can I highlight elements in a matrix?

Question

I'm about to use this diagram to make computing 3x3 determinants clear and easy for my students. I tried to add this picture to my lecture but it's blurred due to the bad quality of the picture. So i want to create exactly the same figure using latex. I've searched and read a few topics about highlighting elements and diagonals in matrix but it does not work in this case. I want to add plus signs and minus signs as in the picture, and highlight elements in the matrice with elements outside the matrice. I appreciate all your help and your advice. Thank you in advance !

Answer 1

An alternative "pure" TikZ solution:

\documentclass[border=1mm,tikz]{standalone}
\usetikzlibrary{calc,matrix}

    \begin{document}
\begin{tikzpicture}[
strip/.style args = {#1/#2}{
    draw=#1, very thin, fill=#1, opacity=0.2,
    text width=#2,% will be calculated in path
    rounded corners,% only if you like them ...
    minimum height=1em,% adjust to desired height,
    inner xsep=-1mm,% adjust to desired length of strip
    inner ysep=2mm,
    yshift=-0.6ex,
    sloped},
                    ]
\matrix (mtrx)  [matrix of math nodes,
                 column sep=1em,   
                 nodes={text height=2ex,text width=2ex}]
{
|[red]|+
    & |[red]|+
          & \color{red}+\color{blue}- 
                & |[blue]|- 
                      & |[blue]|-   \\
a_1 & b_1 & c_1 & a_1 & a_2   \\
a_2 & b_2 & c_2 & a_2 & b_2   \\
a_3 & b_3 & c_3 & a_3 & b_3   \\
};
\draw[thick] (mtrx-2-1.north) -| (mtrx-4-1.south -- (mtrx-4-1.south);
\draw[thick] (mtrx-2-3.north) -| (mtrx-4-3.south -- (mtrx-4-3.south);
\path let   \p1 = (mtrx-4-1.center),
            \p2 = (mtrx-2-4.center),
            \n1 = {veclen(\y2-\y1,\x2-\x1)} in
      (mtrx-4-1.center) -- node[strip=blue/\n1] {} (mtrx-2-3.center)
      (mtrx-4-2.center) -- node[strip=blue/\n1] {} (mtrx-2-4.center)
      (mtrx-4-3.center) -- node[strip=blue/\n1] {} (mtrx-2-5.center)
%
      (mtrx-2-1.center) -- node[strip=red/\n1] {} (mtrx-4-3.center)
      (mtrx-2-2.center) -- node[strip=red/\n1] {} (mtrx-4-4.center)
      (mtrx-2-3.center) -- node[strip=red/\n1] {} (mtrx-4-5.center);
\end{tikzpicture}
    \end{document}

If you like to have different colors, you can defined similarly as this do @MyUserlsThis in his/her answer.

Edit: I wasn't aware for this answer, @egreg thank you for point it. Considered it, the solutions is more simple (and beautiful):

\documentclass[border=1mm,tikz]{standalone}
\usetikzlibrary{calc,matrix}

        \begin{document}
\begin{tikzpicture}[
strip/.style = {
    draw=#1,%color
    line width=1em, opacity=0.2,
    line cap=round ,% only if you like them ...
                            },
                    ]
\matrix (mtrx)  [matrix of math nodes,
                 column sep=1em,   
                 nodes={text height=2ex,text width=2ex}]
{
|[red]|+
    & |[red]|+
          & \color{red}+\color{blue}- 
                & |[blue]|- 
                      & |[blue]|-   \\
a_1 & b_1 & c_1 & a_1 & a_2   \\
a_2 & b_2 & c_2 & a_2 & b_2   \\
a_3 & b_3 & c_3 & a_3 & b_3   \\
};
\draw[thick] (mtrx-2-1.north) -| (mtrx-4-1.south west) 
                              -- (mtrx-4-1.south);
\draw[thick] (mtrx-2-3.north) -| (mtrx-4-3.south east)
                              -- (mtrx-4-3.south);
\path[draw,strip=red,shorten <=-1mm]
    (mtrx-4-1.center) edge (mtrx-2-3.center)
    (mtrx-4-2.center) edge (mtrx-2-4.center)
    (mtrx-4-3.center)  --  (mtrx-2-5.center);
\path[draw,strip=blue,shorten >=-1mm]
    (mtrx-2-1.center) edge (mtrx-4-3.center)
    (mtrx-2-2.center) edge (mtrx-4-4.center)
    (mtrx-2-3.center)  --  (mtrx-4-5.center);

\end{tikzpicture}
    \end{document}

which gives:

Edit (2): Regarding to comment of @BarbaraBenton: signs + and - can be moved nearly line width reducing distance between the first and second row. This and also lookout, when strips is rectangular one can obtain with next MWE (where also colors are interchanged):

\documentclass[border=1mm,tikz]{standalone}
\usetikzlibrary{calc,matrix}

        \begin{document}
\begin{tikzpicture}[
strip/.style = {
    draw=#1,%color
    line width=1em, opacity=0.2,
    shorten <=-2mm,shorten >=-2mm,
                            },
                    ]
\matrix (mtrx)  [matrix of math nodes,
                 column sep=1em,
                 nodes={text height=1ex,text width=2ex}]
{
|[red]|+
    & |[red]|+
          & \color{red}+\color{blue}-
                & |[blue]|-
                      & |[blue]|-   \\[3.3mm,between origins]
a_1 & b_1 & c_1 & a_1 & a_2         \\
a_2 & b_2 & c_2 & a_2 & b_2         \\
a_3 & b_3 & c_3 & a_3 & b_3         \\
};
\draw[thick] (mtrx-2-1.north) -| (mtrx-4-1.south west)
                              -- (mtrx-4-1.south);
\draw[thick] (mtrx-2-3.north) -| (mtrx-4-3.south east)
                              -- (mtrx-4-3.south);
\path[draw,strip=blue]
    (mtrx-4-1.center) edge (mtrx-2-3.center)
    (mtrx-4-2.center) edge (mtrx-2-4.center)
    (mtrx-4-3.center)  --  (mtrx-2-5.center);
\path[draw,strip=red]
    (mtrx-2-1.center) edge (mtrx-4-3.center)
    (mtrx-2-2.center) edge (mtrx-4-4.center)
    (mtrx-2-3.center)  --  (mtrx-4-5.center);
\end{tikzpicture}
    \end{document}

Of course, there is still place to make fine adjustment. For example smaller signs + and -, or different positioning, or etc, but something should be left to OP (if he will accept the answer).

Edit (3): For use with document class beamer, here is complete and many time tested MWE (with recent Beamer: version 3.36):

    \documentclass{beamer} 
\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{calc,matrix}

        \begin{document}
\begin{frame}[fragile]
\begin{tikzpicture}[
strip/.style = {
    draw=#1,%color
    line width=1em, opacity=0.2,
    shorten <=-2mm,shorten >=-2mm,
                            },
                    ]
\matrix (mtrx)  [matrix of math nodes,
                 column sep=1em,
                 nodes={text height=1ex,text width=2ex}]
{
|[red]|+
    & |[red]|+
          & \color{red}+\color{blue}-
                & |[blue]|-
                      & |[blue]|-   \\[3.3mm,between origins]
a_1 & b_1 & c_1 & a_1 & a_2         \\
a_2 & b_2 & c_2 & a_2 & b_2         \\
a_3 & b_3 & c_3 & a_3 & b_3         \\
};
\draw[thick] (mtrx-2-1.north) -| (mtrx-4-1.south west)
                              -- (mtrx-4-1.south);
\draw[thick] (mtrx-2-3.north) -| (mtrx-4-3.south east)
                              -- (mtrx-4-3.south);
\path[draw,strip=blue]
    (mtrx-4-1.center) edge (mtrx-2-3.center)
    (mtrx-4-2.center) edge (mtrx-2-4.center)
    (mtrx-4-3.center)  --  (mtrx-2-5.center);
\path[draw,strip=red]
    (mtrx-2-1.center) edge (mtrx-4-3.center)
    (mtrx-2-2.center) edge (mtrx-4-4.center)
    (mtrx-2-3.center)  --  (mtrx-4-5.center);
\end{tikzpicture}
\end{frame}
    \end{document}

Answer 2

After playing around with tikz, I could get it, I believe, decently. I'm just a tikz beginner, so I am 100% sure this can be done better, without needing so much fine-tuning of positions. You can adjust all positions and colors though:

\documentclass{article}

\usepackage{tikz}
\usepackage{amsmath}
\usepackage{xcolor}
\definecolor{myblue}{rgb}{0,0,0.8}
\definecolor{myred}{rgb}{0.8,0,0}


\begin{document}
\begin{tikzpicture}
\node[fill=none,draw=none] (matrix)
{
\begin{minipage}{\textwidth}
\[ \begin{bmatrix} 
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{bmatrix}
\begin{matrix}
a_1 & b_1 \\
a_2 & b_2 \\
a_3 & b_3
\end{matrix} \]
\end{minipage}
};
\draw[myblue,fill=myblue,opacity=0.2] (-1.4,-0.4)--(0.25,0.6)--(0.4,0.4)--(-1.25,-0.6);
\draw[myblue,fill=myblue,opacity=0.2] (-0.7,-0.4)--(0.9,0.6)--(1.05,0.4)--(-0.55,-0.6);
\draw[myblue,fill=myblue,opacity=0.2] (0,-0.4)--(1.55,0.6)--(1.7,0.4)--(0.15,-0.6);
\draw[myred,fill=myred,opacity=0.2] (-1.4,0.4)--(0.25,-0.6)--(0.4,-0.4)--(-1.25,0.6);
\draw[myred,fill=myred,opacity=0.2] (-0.7,0.4)--(0.85,-0.6)--(1.0,-0.4)--(-0.55,0.6);
\draw[myred,fill=myred,opacity=0.2] (0,0.4)--(1.55,-0.6)--(1.7,-0.4)--(0.15,0.6);
\node[above of=matrix,xshift=-1.3cm,yshift=-0.2cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=-0.6cm,yshift=-0.2cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=-0cm,yshift=-0.2cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=0.4cm,yshift=-0.2cm] {\textcolor{myblue}{$-$}};
\node[above of=matrix,xshift=1cm,yshift=-0.2cm] {\textcolor{myblue}{$-$}};
\node[above of=matrix,xshift=1.6cm,yshift=-0.2cm] {\textcolor{myblue}{$-$}};
\end{tikzpicture}
\end{document}

EDIT: In view of the comment by barbara, I've tried to enlarge the space between rows, to make it a little easier to see. I've done it following exactly this: incrementing matrix space. The only problem is that the matrix's bracket is too tall (it ends up being above the plus and - signs), and I don't know how to fix it. I leave the code and result, and if anyone knows how to do so, I'll happily edit this. Or you can just increase the yshift value of the plus and minus signs to put them above the matrix.

\documentclass{article}

\usepackage{tikz}
\usepackage{amsmath}
\usepackage{xcolor}
\definecolor{myblue}{rgb}{0,0,0.8}
\definecolor{myred}{rgb}{0.8,0,0}

\begingroup
\renewcommand*{\arraystretch}{1.5}
% your pmatrix expression
\endgroup

\makeatletter
\renewcommand*\env@matrix[1][\arraystretch]{%
  \edef\arraystretch{#1}%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{*\c@MaxMatrixCols c}}
\makeatother

\begin{document}
\begin{tikzpicture}
\node[fill=none,draw=none] (matrix)
{
\begin{minipage}{\textwidth}
\[ \begin{bmatrix}[1.5] 
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{bmatrix}
\begin{matrix}[1.5]
a_1 & b_1 \\
a_2 & b_2 \\
a_3 & b_3
\end{matrix} \]
\end{minipage}
};
\draw[myblue,fill=myblue,opacity=0.2] (-1.4,-0.7)--(0.2,0.75)--(0.35,0.55)--(-1.25,-0.9);
\draw[myblue,fill=myblue,opacity=0.2] (-0.7,-0.7)--(0.8,0.75)--(0.95,0.55)--(-0.55,-0.9);
\draw[myblue,fill=myblue,opacity=0.2] (0,-0.7)--(1.45,0.75)--(1.6,0.55)--(0.15,-0.9);
\draw[myred,fill=myred,opacity=0.2] (-1.4,0.55)--(0.25,-0.9)--(0.4,-0.7)--(-1.25,0.75);
\draw[myred,fill=myred,opacity=0.2] (-0.7,0.55)--(0.85,-0.9)--(1.0,-0.7)--(-0.55,0.75);
\draw[myred,fill=myred,opacity=0.2] (0,0.55)--(1.55,-0.9)--(1.7,-0.7)--(0.15,0.75);
\node[above of=matrix,xshift=-1.3cm,yshift=-0.1cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=-0.6cm,yshift=-0.1cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=-0cm,yshift=-0.1cm] {\textcolor{myred}{$+$}};
\node[above of=matrix,xshift=0.4cm,yshift=-0.1cm] {\textcolor{myblue}{$-$}};
\node[above of=matrix,xshift=1cm,yshift=-0.1cm] {\textcolor{myblue}{$-$}};
\node[above of=matrix,xshift=1.6cm,yshift=-0.1cm] {\textcolor{myblue}{$-$}};
\end{tikzpicture}
\end{document}

Answer 3

With {NiceMatrix} of nicematrix (≥ 5.16 2021-06-20) and Tikz to draw the rules.

\documentclass{article}
\usepackage{nicematrix}
\usepackage{tikz}
\begin{document}
$\begin{NiceMatrix}[columns-width=7mm]
\CodeBefore [create-cell-nodes]
    \begin{tikzpicture} 
      [line width=3.5mm, opacity=0.2, shorten < = -3mm, shorten > = -3mm]
      \draw [blue] (4-1.center) -- (2-3.center) ;
      \draw [blue] (4-2.center) -- (2-4.center) ;
      \draw [blue] (4-3.center) -- (2-5.center) ;
      \draw [red] (2-1.center) -- (4-3.center) ;
      \draw [red] (2-2.center) -- (4-4.center) ;
      \draw [red] (2-3.center) -- (4-5.center) ;
    \end{tikzpicture}
\Body
    \color{red} +   & \color{red} +   & \color{red}+\color{blue}-  & \color{blue}-   & \color{blue}- \
    a_1 & b_1 & c_1 & a_1 & a_2   \[2mm]
    a_2 & b_2 & c_2 & a_2 & b_2   \[2mm]
    a_3 & b_3 & c_3 & a_3 & b_3   \
\CodeAfter
    \SubMatrix[{2-1}{4-3}]
\end{NiceMatrix}$
\end{document}

You need several compilations (because of the PGF/Tikz nodes).

Answer 4

Combining

• the \bgar macro that puts an arrow in the background and
• the lr delim style that puts delimiters around arbitrary submatrices

we can achieve this with TikZ-CD without having to draw over nodes.

Code

\documentclass{article}
\usepackage{tikz-cd,amsmath}
\usetikzlibrary{backgrounds,calc}
\tikzcdset{
  math matrix/.style={
    column sep=1.5*\the\arraycolsep, row sep=.3*\the\baselineskip,
    cells={nodes={font=\mathstrut, inner sep=+0pt, outer sep=+0pt}}},
  lr delim/.style={/tikz/every matrix/.append style={lr delim={#1}}}}
\tikzset{% https://tex.stackexchange.com/a/660470/16595
  lr delim/.style args={#1#2 and #3#4 around #5 to #6}{append after command={
    {{[local bounding box=@] (\tikzlastnode-#5.north west)
     (\tikzlastnode-#6.south east)}(@)[every left delimiter/.append style={#2},
                                      every right delimiter/.append style={#4},
                 late options={left delimiter={#1},right delimiter={#3}}]}[]}}}
\makeatletter
\let\tikzcd@ar@new@orig\tikzcd@ar@new
\def\tikzcd@ar@new@bg[#1]{% https://tex.stackexchange.com/a/664328/16595
  \pgfutil@g@addto@macro\tikzcd@savedpaths{%
    \pgfonlayer{background}\path[/tikz/commutative diagrams/.cd,every arrow,#1]%
    (\tikzcd@ar@start\tikzcd@startanchor)to(\tikzcd@ar@target\tikzcd@endanchor);
    \endpgfonlayer}\let\tikzcd@ar@new\tikzcd@ar@new@orig}
\tikzcdset{every diagram/.append code=%
  \def\bgar{\let\tikzcd@ar@new\tikzcd@ar@new@bg\tikzcd@arrow}}
\makeatother
\tikzcdset{
  highlight/.style 2 args={
    /tikz/arrows=-, draw opacity=.5, draw=#1!50, text=#1, line width=.95em,
    line cap=round, to path={let\p0=($(\tikztostart)-(\tikztotarget)$),
      \n0={atan2(\y0,\x0)} in (\tikztostart.\n0)--(\tikztotarget.\n0-180)
        node[pos=1.2]{$#2$}\tikztonodes}}}
\begin{document}
\[\begin{tikzcd}[math matrix, lr delim={[ and ] around 1-1 to 3-3},
  cells={nodes={align=center,text width=width("$a_0$")}},
  minus/.style={highlight={blue}{-}}, plus/.style={highlight={red}{+}}]
  a_1 & b_1 & c_1 & a_1 & b_1 \\
  a_2 & b_2 & c_2 & a_2 & b_2 \\
  a_3 \bgar[rruu, minus]
& b_3 \bgar[rruu, minus]
& c_3 \bgar[rruu, minus] \bgar[lluu, plus]
& a_3                    \bgar[lluu, plus]
& b_3                    \bgar[lluu, plus]
\end{tikzcd}\]
\newcommand\mPlus[1] {\mathcolor{red} {#1}}
\newcommand\mMinus[1]{\mathcolor{blue}{#1}}
[ \det A = (\mPlus {a_1b_2c_3} + \mPlus {b_1c_2a_3} + \mPlus {c_1a_2b_3})
           -(\mMinus{a_3b_2c_1} + \mMinus{b_3c_2a_1} + \mMinus{c_3a_2b_1}) ]
\end{document}

Output