# 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.

### Addition of matrices

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 .