How to build a table of fixed width and fixed height? (Edit: the question left now is about color overwrite)

Question

\begin{frame}{Title}
\begin{equation*}
            \begin{tabular}{cccc}

                  & aaaaaaaaaa & aaaaaaaaa & aaaaaaaaaaaaa \\ \cline{2-4}
                $A_A \prec_B C_C$ & \multicolumn{1}{|c}{} 
                    & \multicolumn{1}{|c}{}   
                        &
                    \multicolumn{1}{|c|}{}   
                \\ \cline{2-4}
                $A_A {\tiny\cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B} C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                    & \multicolumn{1}{|c}{} 
                        & \multicolumn{1}{|c|}{} 
                     
                \\ \cline{2-4}
                $A_A=C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                    & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                        & \multicolumn{1}{|c|}{} 
                     
                \\ \cline{2-4}
                $A_A \succ_B C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                    & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                        & \multicolumn{1}{|c|}{} 
                \\ \cline{2-4}
                
            \end{tabular}
\end{equation*}
\end{frame}

(Deleted)The problem of width is caused by each column having different numbers of "a". And the problem of height is caused by a customized large symbol. I tried to replace the initial four "c" with "p{4cm}", but it does not respond at all. (this is for adjusting the width. As for height, shall I use something like "\renewcommand{\arraystretch}{1.5}"?) I have searched some threads, but their table starts with format like {|c|c|} and I feel that my case does not fit in?

There is also another problem: some lines are covered by the gray block. I tried to move \cline before \ rather than after \, but it always says "error". There are some relevant thread: Cellcolor overwrites partial horizontal lines (\cline) But their tables start with something like {|c|c|}, so all the border lines are present, which is inconsistent with my case.

Thank you very much!

Edit(delted):

\begin{frame}{Frame Title}
\noindent\begin{minipage}[t][6in][t]{\textwidth}
\centering
        \begin{tabular}{@{} c p{1in} p{1in} p{1in} @{}}

              & aaaaaaaaaa & aaaaaaaaa & aaaaaaaaaaaaa \\ \cline{2-4}
            $A_A \prec_B C_C$ & \multicolumn{1}{|c}{} 
                & \multicolumn{1}{|c}{}   
                    &
                \multicolumn{1}{|c|}{}   
            \\ \cline{2-4}
            $A_A \cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                & \multicolumn{1}{|c}{} 
                    & \multicolumn{1}{|c|}{} 
                 
            \\ \cline{2-4}
            $A_A=C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                    & \multicolumn{1}{|c|}{} 
                 
            \\ \cline{2-4}
            $A_A \succ_B C_C$ & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                & \multicolumn{1}{|c}{\cellcolor[HTML]{C0C0C0}} 
                    & \multicolumn{1}{|c|}{} 
            \\ \cline{2-4}
            
        \end{tabular}
\end{minipage}

\end{frame}

Answer 1

Here is a solution with {NiceTabular} of nicematrix.

• The key columns-width=auto sets the width of all the columns to the largest one.

• I have set \arraystretch to 1.4, which, in your case, is enough to have all the rows of the same height.

• The command \Block[hvlines]{*-*}{} creates a block with no content (the last argument) but which is used to have all the rules drawn in that block (with the key hvlines).

• With {NiceTabular} of nicematrix, the rules are drawn by PGF (a sublayer of Tikz) after the construction of the array and that ensure a perfect result with the colored cells, in all the PDF viewers). However, you need several compilations.

\documentclass{beamer}
\usepackage{cancel}
\usepackage{nicematrix}

\begin{document}

\begin{frame}{Title}
\renewcommand{\arraystretch}{1.4}
\begin{NiceTabular}{cccc}[colortbl-like,columns-width=auto]
& aaaaaaaaaa & aaaaaaaaa & aaaaaaaaaaaaa \\ 
$A_A \prec_B C_C$ & \Block[hvlines]{*-*}{} \\
$A_A {\tiny\cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B} C_C$ & \cellcolor{gray!50} \\
$A_A=C_C$ & \cellcolor{gray!50} & \cellcolor{gray!50} \\
$A_A \succ_B C_C$ & \cellcolor{gray!50} & \cellcolor{gray!50} \\
\end{NiceTabular}
\end{frame}

\end{document}

Answer 2

I would wipe out all lines -- horizontal & vertical -- within the shaded area.

\documentclass[table]{beamer}
\usepackage{cancel,xcolor,tabularx,ragged2e}
\newcolumntype{C}{>{\Centering}X}
\newcommand\cc{\multicolumn{1}{c}{\cellcolor[HTML]{C0C0C0}}} % handy shortcut macro

\begin{document}
\begin{frame}{Title}
\begin{tabularx}{\textwidth}{l|*{3}{C|}}
\cline{2-4}
& aaaaaaaaaa & aaaaaaaaa & aaaaaaaaaaaaa \\ 
\cline{2-4}
$A_A \prec_B C_C$ 
   & & & \\ \cline{2-4}
$A_A \mathrel{{\scriptscriptstyle\cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B}} C_C$ 
   & \cc &     & \\ \cline{2-4}
$A_A=C_C$ 
   & \cc & \cc & \\ \cline{2-4}
$A_A \succ_B C_C$ 
   & \cc & \cc & \\ \cline{2-4}        
\end{tabularx}
\end{frame}
\end{document}

Answer 3

With use of the scalebox (defined in the adjustbox package) and tabularay package:

\documentclass{beamer}
\usepackage{adjustbox}
\usepackage{cancel}
\usepackage{xcolor}
\usepackage{tabularray} % tested with version 2021P

\begin{document}

\begin{frame}
\frametitle{Title}
    \begin{tblr}{colspec={Q[r,$] *3{X[c]}}, % or Q[r,mode=math]
                 hline{2-Z}={2-Z}{solid},
                 vline{2-Z}={2-Z}{solid},
                 row{2-Z} = {ht=4ex}
                }
                & aaaaaaaaaa            & aaaaaaaaa         & aaaaaaaaaaaaa \\
A_A \prec_B C_C &                       &                   &               \\
A_A  \scalebox{0.7}{$\cancel{\overset{\prec}{\underset{\succ}{\approx}}}$}_B C_C
                & \SetCell{gray!50}     &                   &               \\
A_A=C_C         & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
A_A \succ_B C_C & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
    \end{tblr}
\end{frame}

\end{document}

Edit (1): Now columns 2 - 4 have equal width (and height).

Edit (2): If you use Overleaf, you need to load/install the package tabularrray and ninecolors (which is requiewd by taabularray) yourself. Since I'm not familiar with Overleaf, I can't advice you, how to do this, but their support is very friendly and and they will be happy to help you with that.

Also you not need to use adjustbox package if for math expression in the second row use suggestion in the @Mico answer:

\documentclass{beamer}
\usepackage{cancel}
\usepackage{xcolor}
\usepackage{tabularray} % tested with version 2021P

\begin{document}

\begin{frame}
\frametitle{Title}
    \begin{tblr}{colspec={Q[r,$] *3{X[c]}}, % or Q[r,mode=math]
                 hline{2-Z}={2-Z}{solid},
                 vline{2-Z}={2-Z}{solid},
                 row{2-Z} = {ht=4ex}
                }
                & aaaaaaaaaa            & aaaaaaaaa         & aaaaaaaaaaaaa \\
A_A \prec_B C_C &                       &                   &               \\
A_A \mathrel{\scriptstyle\cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B} C_C % adopted from @Mico suggestion
                & \SetCell{gray!50}     &                   &               \\
A_A=C_C         & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
A_A \succ_B C_C & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
    \end{tblr}
\end{frame}

\end{document}

Addendum: Regarding your comment:

• if you like to add some (complete) rows below existing, then just write it, as I did in MWE below (I don't know what is content of this row)
• if you only like have multiline content of particular cell, than yo add this text to this cell:
  • in case that text is long enough, it will break in X column types into two lines automatically, for example new text aaaaaaaaa
  • if it is shorter, than you need to break it manually, for example new\\ aaaaaaaaa

\documentclass{beamer}
\usepackage{cancel}
\usepackage{xcolor}
\usepackage{tabularray} % tested with version 2021P

\begin{document}

\begin{frame}
\frametitle{Title}
    \begin{tblr}{colspec={Q[r,$] *3{X[c,b]}}, % or Q[r,mode=math]
                 hline{2-Z}={2-Z}{solid},
                 vline{2-Z}={2-Z}{solid},
                 row{2-Z} = {ht=4ex}
                }
                & aaaaaaaaaa            & new text 
                                          aaaaaaaaa         
                                                            & aaaaaaaaaaaaa \\
A_A \prec_B C_C &                       &                   &               \\
A_A \mathrel{\scriptstyle\cancel{\overset{\prec}{\underset{\succ}{\approx}}}_B} C_C % adopted from @Mico suggestion
                & \SetCell{gray!50}     &                   &               \\
A_A=C_C         & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
A_A \succ_B C_C & \SetCell{gray!50}     & \SetCell{gray!50} &               \\
A_A \nprec_B C_C    & \SetCell{gray!50} & \SetCell{gray!50} & \SetCell{gray!50}              \\
    \end{tblr}
\end{frame}

I tested all the examples with the latest MiKTEX program. The compilation was without warnings or errors. If you use Overleaf (as I concluded from one of your comment to other answers) then you may have problems with compilation. In this case please contact their support team and ask for help.

I allowed myself to add an overleaf tag to your question that you will get their attention.