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:

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

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:

$(AB)_{ij}=\sum _{k}A_{ik}B_{kj}$

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB \neq BA} .

The $n\times n$ identity matrix $I_{n}$ satisfies the property:

AI_{n}=I_{n}A=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=I_{n}

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 $GL_{n}(\mathbb {R} )$ .

Matrix concepts

Basic concepts

Advanced concepts