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\begin{center}\begin{large} Arrays and Matrices \\
\end{large}       Matchett, \today
\end{center}

Arrays are produced in math mode with the \emph{array} command.  
An array with 4 rows and 3 columns is produced by the following 
syntax: 
 \begin{verbatim}
 \begin{array}{***}  row1 \\ row2 \\ row3 \\ row4  \end{array} 
 \end{verbatim}
Here each * is one of the letters, c, l, or r, and each row
consists of 3 entries separated by ampersands.  Note that each 
row ends with a double backslash except for the last row which 
simply ends with the command that ends the environment.
Here is a comprehensive example.  If you put the lines,
  \begin{verbatim} 
  \begin{displaymath}
     \begin{array}{ccr} 271 & 828 & 182 \\ 4 & 5 & 9 \\ 
      0 & 8+x & 2 \\ 1 & 2 & 3   \end{array}
  \end{displaymath}
  \end{verbatim}
in a LaTeX file, the pdf file will have the display,
  \begin{displaymath}
     \begin{array}{ccr} 271 & 828 & 182 \\ 4 & 5 & 9 \\ 
      0 & 8+x & 2 \\ 1 & 2 & 3   \end{array}
  \end{displaymath}
Which of the three letters l, r, and c is used for a particular 
column determines whether the column is left justified, right 
justified, or centered.  

We often want to enclose a matrix in large brackets.  This is 
accomplished by the commands $\backslash$left[ and $\backslash$right].  
One could use parentheses, curley braces, or vertical lines as large 
delimiters here instead of brackets.  So the following syntax:
 \begin{verbatim}
    \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)
 \end{verbatim}
produces
 \begin{displaymath}
     \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)
 \end{displaymath}
Remember that a curley brace that is to be printed in the pdf 
file must be preceeded by a backslash in the LaTeX file.  LaTeX 
keeps track of ``left'' and ``right'' delimiters, and requires 
that they come in left-right pairs.  However, as the above 
example shows, a pair can be mismatched as long as there is a 
left one and a right one.  LaTeX treats an array as a single 
large character.  So arrays may be easily used in equations, like 
 \begin{displaymath}  
   A = \left[ \begin{array}{cc} 8 & 9 \\ u & v \end{array} \right]
 \end{displaymath}
If it is desired to have a large delimiter on only one side of an 
array, that delimiter can be paired with a period on the other 
side.  Here is an example of that.  The latex expression, 
 \begin{verbatim} 
   f(x) = \left\{ \begin{array}{rl} 
                 2x, &  x \le 1 \\ 3-x, & x > 1 
          \end{array} \right.
 \end{verbatim}
produces 
 \begin{displaymath}
   f(x) = \left\{ \begin{array}{rl} 
                 2x, &  x \le 1 \\ 3-x, & x > 1 
          \end{array} \right.
 \end{displaymath}
in the pdf file.


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