Horizontal stretching of pmatrix

Use \arraycolsep:

\documentclass{article}
\usepackage[utf8]{inputenc}

\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{graphicx}


\begin{document}

\setlength{\arraycolsep}{10pt}.
\renewcommand\arraystretch{2}

\[
\begin{pmatrix}
  2 & 1 & 0 & 0 & 1 & 0 \\
  1 & 0 & 1 & 0 & 1 & 0 \\
  0 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 1 & 0 & 1 & 1 \\
  1 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0
\end{pmatrix}
\]

\end{document}

And also please un-learn $$, it's deprecated.

You can play with \arraycolsep, and to ensure a proper spacing between the contents of the matrix and the parentheses, you can nest a matrix environment, surrounded by a pair of horizontal spaces, in a pmatrix environment, like this:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}

\begin{document}

\[ \setlength{\arraycolsep}{10pt}
\renewcommand\arraystretch{2}
  \begin{pmatrix}
    \hspace {0.7em}
  \begin{matrix}
  2 & 1 & 0 & 0 & 1 & 0 \\
  1 & 0 & 1 & 0 & 1 & 0 \\
  0 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 1 & 0 & 1 & 1 \\
  1 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 &0 & 0
  \end{matrix}
    \hspace {0.7em}
  \end{pmatrix}
\]

\end{document} 

One possibility is to use tabular*. Please see here for more options.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\renewcommand\arraystretch{2}

\begin{document}

\[
 \left(\vcenter{\hbox{\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}} *{6}{c}}
  2 & 1 & 0 & 0 & 1 & 0 \\
  1 & 0 & 1 & 0 & 1 & 0 \\
  0 & 1 & 0 & 1 & 0 & 0  \\
  0 & 0 & 1 & 0 & 1 & 1 \\
  1 & 1 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 & 0
 \end{tabular*}}}\right)
\]

\end{document}