# 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:

$\begin{bmatrix} x & y & z \\ 1 & 3 & 5 \\ 0 & 2 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 3 & 1 \\ 4 & 3 & {x+2} \\ 0 & 4 & v \end{bmatrix}$

would equal

$\begin{bmatrix} {x+0} & {y+3} & {z+1} \\ {1+4} & {3+3} & {5+(x+2)} \\ {0+0} & {2+4} & {0+v} \end{bmatrix} = \begin{bmatrix} x & {y+3} & {z+1} \\ 5 & 6 & {x+7} \\ 0 & 6 & v \end{bmatrix}$

### Multiplication of matrices

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

 (AB)ij = ∑ AikBkj k

However, not every pair of matrices can be multiplied. In order for matrices A and B to be compatible for multiplication, the number of columns in A must equal the number of rows in B. If A is an $m \times n$ matrix and B is an $n \times p$ matrix, the product matrix AB will have m rows and p columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for $AB \neq BA$.

The $n \times n$ identity matrix In satisfies the property:

AIn = InA = A

for all $n \times n$ matrices A.

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

AB = BA = In

This means that for every positive integer n, the set of all $n \times n$ matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group $GL_{n}(\mathbb{R})$.