# Matrices

A
**
matrix
**
is a rectangular array of numbers enclosed by brackets. (The plural of matrix is
**
matrices.
**
)

$\begin{array}{rrrr}\hfill \left[\begin{array}{rr}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\\ \hfill 7& \hfill -1\end{array}\right]& \hfill \left[\begin{array}{rrr}\hfill 6& \hfill -2& \hfill -1\end{array}\right]& \hfill \left[\begin{array}{r}\hfill -5\\ \hfill 3\\ \hfill 10\end{array}\right]& \hfill \left[\begin{array}{rr}\hfill 1& \hfill -1\\ \hfill 3& \hfill -9\end{array}\right]\end{array}$ are all examples of matrices.

The numbers in a matrix are called the
**
elements
**
(or
**
entries)
**
of the matrix. The number of
**
rows
**
(horizontal) and the number of
**
columns
**
(vertical) determine the
dimensions of the matrix
. You always write the number of rows first and the number of columns second. In order, the dimensions of the above matrices are
$3\times 2$
(read
$3$
by
$2$
),
$1\times 4$
,
$3\times 1$

and
$2\times 2$
.

A matrix with only one row (the second one above) is called a
**
row matrix.
**
If the matrix has only one column (the third one above) is a
**
column matrix.
**
The last matrix above is a
square matrix
because the number of rows equals the number of columns.

If all of the elements of a matrix are zero, it is called a
**
zero matrix
**
.

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$ is a $2\times 3$ zero matrix, denoted ${0}_{2\times 3}$ .

One common use of matrices is for solving systems of linear equations . For this, you need to know about matrix row operations and the identity matrix .

You can also do algebra with matrices -- that is, you can
add them and subtract them
,
multiply
them (if their dimensions are compatible), and even do a sort of division by finding their
inverses
(this only works for square matrices). In advanced mathematics, matrices are used to describe
**
linear transformations
**
.