Draw a line through one column of a matrix

\documentclass{article} 
\usepackage{amsmath}
\begin{document}

\[\begin{pmatrix}
  a & b & c  \\
  d & e & f\makebox(-6,0){\rule[1ex]{0.4pt}{3\normalbaselineskip}}\\
  g & h & i
\end{pmatrix}\]

\end{document}

And a line only through f and i:

d & e & f\makebox(-6,0){\rule[-7ex]{0.4pt}{2\normalbaselineskip}}\\

The usual TikZ answer using \tikzmark to mark the top and bottom points where you want the line to be drawn:

Notes:

• This does require two runs. First one to determine the locations, and the second to do the drawing.

References:

• \tikzmark is from Adding a large brace next to a body of text.

Code:

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}

\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline] \node [anchor=base] (#1) {$#2$};}

\newcommand{\DrawLine}[3][]{%
  \begin{tikzpicture}[overlay,remember picture]
    \draw[#1] (#2.north) -- (#3.south);
  \end{tikzpicture}
}

\begin{document}
\[
\begin{pmatrix}
  \tikzmark{topA}{a} & \tikzmark{topB}b & {0} \\
  a & b & 0 \\
  a & b & 0 \\
  \tikzmark{bottomA}{c} & \tikzmark{bottomB}d & {1}
\end{pmatrix}
\]
\DrawLine[red, thick, opacity=0.5]{topA}{bottomA}
\end{document}

\documentclass[a4paper]{article}
\usepackage{amsmath}
\begin{document}
\[
\settodepth{\dimen0}{$\begin{array}[t]{l}1\\2\\3\\4\end{array}$}
\addtolength{\dimen0}{2ex}
\begin{pmatrix}
  a & b & 0 \\
  a & b & 0 \\
  a & b & 0 \\
  \smash{\makebox[0pt][l]{\kern2pt\rule[-.8ex]{0.4pt}{\dimen0}}}
  c & d & 1
\end{pmatrix}
\]
\end{document}

More can be added if you define better your needs