How do I typeset vertical and horizontal lines inside a matrix?

You can use rules to create the vertical and horizontal bars and then just put them into your array directy.

\documentclass{article}
\usepackage{array}
\begin{document}
\newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}}
\newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}}

\[
A = 
\left[
  \begin{array}{cccc}
    \vertbar & \vertbar &        & \vertbar \\
    a_{1}    & a_{2}    & \ldots & a_{n}    \\
    \vertbar & \vertbar &        & \vertbar 
  \end{array}
\right]
\]
\setlength{\extrarowheight}{1ex}
\[
A =
\left[
  \begin{array}{ccc}
    \horzbar & a^{T}_{1} & \horzbar \\
    \horzbar & a^{T}_{2} & \horzbar \\
             & \vdots    &          \\
    \horzbar & a^{T}_{n} & \horzbar
  \end{array}
\right]
\]

\end{document}

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

\[
A = 
\left[
  \begin{array}{cccc}
    \vrule & \vrule & & \vrule\\
    a_{1} & a_{2} & \ldots & a_{n} \\
    \vrule & \vrule & & \vrule 
  \end{array}
\right]
\]
\setlength{\extrarowheight}{1ex}
\[
A =
\left[
  \begin{tabular}{c>{$}c<{$}c}
    --- & a^{T}_{1} & ---\\
    --- & a^{T}_{2} & ---\\
    & \vdots & \\
    --- & a^{T}_{n} & ---
  \end{tabular}
\right]
\]

\end{document}