# 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 $3×2$ (read $3$ by $2$ ), $1×4$ , $3×1$
and $2×2$ .

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 $2×3$ zero matrix, denoted ${0}_{2×3}$ .

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 .