# Matrix

For the 1999 film, see The Matrix.

A matrix (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of numerals. In mathematics, a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level algebra. Matrix algebra is usually taught in sophomore high school level mathematics.

More formally, a matrix is an example of a rank-2 tensor.

Alternately, a matrix may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the synergy of the networked objects.

## Mathematics

In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.

To add two matrices, one would add their respective elements. For example:



would equal



### Multiplication of matrices

To multiply two matrices, use the rule for finding the product of two matrices:



However, not every pair of matrices can be multiplied. In order for matrices  and  to be compatible for multiplication, the number of columns in  must equal the number of rows in . If  is an  matrix and  is an  matrix, the product matrix  will have  rows and  columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for .

The  identity matrix  satisfies the property:



for all  matrices .

Moreover, every square matrix  with nonzero determinant has an inverse matrix  such that



This means that for every positive integer , the set of all  matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group .