In mathematics, a

**matrix** (plural

**matrices**) is a rectangular table of

*elements* (or

*entries*), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices are described by the field of matrix theory. They can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra.

**Mathematical definition:**
An

matrix

is a function

where

is any non-empty set.

is the Cartesian product of sets

and

We say that matrix

is a matrix over the set

. Important thing to note is that, if we want to have matrix algebra, the set

must be a ring and matrix

must be a square matrix (see Square matrices and related definitions below for further explanation). Since the set of all square matrices over a ring is also a ring, matrix algebra is usually called matrix ring.

Since this article mainly considers matrices over real numbers, matrices shown here are actually functions

** For Example**
The matrix

or

is a

matrix. The element

*a*2,3 or

is 7. In terms of the mathematical definition given above, this matrix is a function

and, for example,

and

The matrix

is a

matrix, or 9-element row vector.

**Square matrices:**
A

**square matrix** is a matrix which has the same number of rows and columns.