A matrix (plural: matrices) is simply a rectangular array of ``things''. For now, we'll assume the ``things'' are numbers, but as you go on in mathematics, you'll find that matrices can be arrays of very general objects. Pretty much all that's required is that you be able to add, subtract, and multiply the ``things''.
Here are some examples of matrices. Notice that it is sometimes useful to have variables as entries, as long as the variables represent the same sorts of ``things'' as appear in the other slots. In our examples, we'll always assume that all the slots are filled with numbers. All our examples contain only real numbers, but matrices of complex numbers are very common.
The first example is a square 
matrix; the next is a
matrix (2 rows and 4 columns--if we talk about a matrix
that is ``
'' we mean it has 
rows and 
columns). The
final two examples consist of a single column matrix, and a single row
matrix. These final two examples are often called ``vectors''--the
first is called a ``column vector'' and the second, a ``row vector''.
We'll use only column vectors in this introduction.
Often we are interested in representing a general matrix
with variables in every location, and that is usually done as follows:
The number in row 
and column 
is represented by 
,
where
and
. Sometimes when there
is no question about the dimensions of a matrix, the entire
matrix can simply be referred to as:
