Creating table using TikZ matrix

Question

I'm trying to create a table in TikZ with

1. the first column right-aligned, and the texts in each cell on the same line
2. the rest of the columns center-aligned, and each has width of 1cm

My attempt is as follows:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix}

\begin{document}

\begin{tikzpicture}
  \matrix(dict)[matrix of nodes,below=of game,
    nodes={align=center,text width=1cm},
    row 1/.style={anchor=south},
    column 1/.style={text width=2cm,align=right},
  ]{
    meaning & $b_1$ & $b_2$ & $b_3$ & $b_1b_2$ & $b_1b_3$ & $b_2b_3$ & $b_1b_2b_3$\\
    common list \\
    private list\\
  };
  \draw(dict-1-1.south west)--(dict-1-8.south east);
  \draw(dict-1-1.north east)--(dict-3-1.south east);
\end{tikzpicture}

\end{document}

The output is not as desired, since the texts in cells dict-2-1 and dict-3-1 spread over two lines. Also, texts in column 1 is not right aligned, despite column 1/.style being specified after node/.style.

What should I do?

Answer 1

Remarks

This is just @Qrrbrbirlbel's comment converted to an answer.

Implementation

\documentclass[tikz]{standalone}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
    \matrix(dict)[matrix of nodes,%below=of game,
        nodes={align=center,text width=1cm},
        row 1/.style={anchor=south},
        column 1/.style={nodes={text width=2cm,align=right}}
    ]{
        meaning & $b_1$ & $b_2$ & $b_3$ & $b_1b_2$ & $b_1b_3$ & $b_2b_3$ & $b_1b_2b_3$\\
        common list \\
        private list\\
    };
    \draw(dict-1-1.south west)--(dict-1-8.south east);
    \draw(dict-1-1.north east)--(dict-3-1.south east);
\end{tikzpicture}
\end{document}

Output