Comma delimiter in pmatrix environment (for row vectors)

Question

Is there a way to have a pmatrix environment from amsmath use non-whitespace delimiters such as a comma? I am mainly asking, because I would like to make a row vector this way. I take it, there are essentially two ways of doing something like that:

\documentclass{standalone}
\usepackage{mathtools}

\begin{document}
\begin{gather*}
    a = \left(1,2,3\right) \\
    b = \begin{pmatrix} 1, & 2, & 3 \end{pmatrix}
\end{gather*}
\end{document}

Now, especially the second part does not look too pretty:

Answer 1

https://tex.stackexchange.com/a/53961/6993 provides one solution. It also reminded me of \DeclarePairedDelimiter from mathtools:

\documentclass{standalone}
\usepackage{mathtools}
\DeclarePairedDelimiter{\Vector}{\lparen}{\rparen}

\begin{document}
\begin{gather*}
    a = \left(1,2,3\right) \\
    b = \begin{pmatrix} 1, & 2, & 3 \end{pmatrix}
    c = \Vector{1,2,3}
\end{gather*}
\end{document}

A starred version \Vector* would precede \lparen and \rparen by \left and \right, respectively.