Create a math matrix in TikZ partition lines

Not to go entirely upstream, but there are also some non-TikZ options using the kbordermatrix and blkarray packages:

\documentclass{article}
\usepackage{kbordermatrix}% http://www.hss.caltech.edu/~kcb/TeX/kbordermatrix.sty
\usepackage{blkarray}% http://ctan.org/pkg/blkarray
\begin{document}

\[
  \kbordermatrix{%
      & x & y & z & w & r & & M \\
    & 1 & 2 & 1 & 0 & 0 & \vrule & 10 \\
    & 3 & 2 & 0 & 1 & 0 & \vrule & 20 \\ \cline{2-8}
    & -2 & -10 & 0 & 0 & 1 & \vrule & 0
  }
​\]

\[
  \begin{blockarray}{crrrrrrc}
    & x & y & z & w & r & M &\\
    \begin{block}{[crrrrr|rc]}
      & 1 & 2 & 1 & 0 & 0 & 10 & \\
      & 3 & 2 & 0 & 1 & 0 & 20 & \\ \BAhhline{~------~}
      & -2 & -10 & 0 & 0 & 1 & 0 & \\
    \end{block}
  \end{blockarray}
\]
\end{document}

The code can be modified to accommodate your needs to improve alignment and spacing. In both examples you can add row indices as well.

A basic approach:

Once you named you matrix (mymatrix in this case), you can refer to it in the mymatrix-row-column notation. Usual positioning such as above=of will also work.

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{matrix,positioning}
\begin{document}
 \begin{tikzpicture}[node distance=-1ex]
  \matrix (mymatrix) [matrix of math nodes,left delimiter={[},right
delimiter={]}]
  {
    1  & 2   & 1 & 0 & 0 & 10 \\
    3  & 2   & 0 & 1 & 0 & 20 \\
    -2 & -10 & 0 & 0 & 1 & 0  \\
  };
\draw[red] (mymatrix-2-1.south west) -- (mymatrix-2-6.south east);
\draw[red] (mymatrix-1-5.north east) -- (mymatrix-3-5.south east);
\node [above=of mymatrix-1-1.north] {x};
\node [above=of mymatrix-1-2.north] {y};
\node [above=of mymatrix-1-3.north] {z};
\node [above=of mymatrix-1-4.north] {w};
\node [above=of mymatrix-1-5.north] {r};
\node [above=of mymatrix-1-6.north] {M};
\end{tikzpicture}

\end{document}