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:

<math>\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} </math>

would equal

<math>\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} </math>

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 <math>A</math> and <math>B</math> to be compatible for multiplication, the number of columns in <math>A</math> must equal the number of rows in <math>B</math>. If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for <math>AB \neq BA</math>.

The <math>n \times n</math> identity matrix <math>I_n</math> satisfies the property:

for all <math>n \times n</math> matrices <math>A</math>.

Moreover, every square matrix <math>A</math> with nonzero determinant has an inverse matrix <math>B</math> such that

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

Matrix concepts

Basic concepts

Advanced concepts