Horizontal Line in Array

You're missing the vertical line | in the column specification and a \multicolumn{1}{r} for the element in the top left corner:

\documentclass{article}

\begin{document}

\[
  \left(
    \begin{array}{ r | *{3}{r} }
      \multicolumn{1}{r}{2} & 2 & -1 &  3 \\
      \cline{2-4}
      0 & 1 &  3 &  4 \\
      0 & 1 &  3 &  4 \\
      0 & 3 &  9 &  6 \\
      0 & 1 &  3 & -2  
    \end{array}
  \right)
\]

\end{document}

For a more evenly distributed spacing around the last three columns, consider using \phantom{-}2 in the first row:

\documentclass{article}

\begin{document}

\[
  \left(
    \begin{array}{ r | *{3}{r} }
      \multicolumn{1}{r}{2} & \phantom{-}2 & -1 &  3 \\
      \cline{2-4}
      0 & 1 &  3 &  4 \\
      0 & 1 &  3 &  4 \\
      0 & 3 &  9 &  6 \\
      0 & 1 &  3 & -2  
    \end{array}
  \right)
\]

\end{document}

With nicematrix. The environments of nicematrix create PGF/Tikz nodes under the rows, columns and cells and the array and it's possible to use them to draw whatever you want with Tikz.

\documentclass{article}
\usepackage{nicematrix}
\usepackage{tikz}

\begin{document}

\[
\begin{pNiceMatrix}[r,right-margin,columns-width=auto]
  2 & 2 & -1 &  3 \\
  0 & 1 &  3 &  4 \\
  0 & 1 &  3 &  4 \\
  0 & 3 &  9 &  6 \\
  0 & 1 &  3 & -2  
\CodeAfter
\tikz \draw ([xshift=4pt]row-6-|col-2) |- (row-2-|col-5) ;
\end{pNiceMatrix}
\]

\end{document}