*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 .

### Matrix concepts

#### Basic concepts

- Adjoint
- Determinant
- Diagonal matrix
- Identity matrix
- Inverse matrix
- Null, column and row space
- Scalar
- Trace
- Transpose matrix
- Vector
- Zero matrix

#### Advanced concepts

- Basis
- Diagonalizable
- Eigenspace
- Eigenvalue
- Eigenvector
- Gram-Schmidt process
- Hermitian matrix
- Jordan canonical form
- Laplacian
- Linear independence
- Matrix diagonalization
- Matrix reformation
- Matrix transformation
- Orthogonal matrix
- Orthonormal matrix
- Resultant
- Span
- Linear equations
- Transcriptor
- Wronskian