Matrix

From Conservapedia

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.

1 Mathematics

Mathematics[edit]

In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.

Addition of matrices[edit]

To add two matrices, one would add their respective elements. For example:

$\text{[math]}$

would equal

$\text{[math]}$

Multiplication of matrices[edit]

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

$\text{[math]}$

However, not every pair of matrices can be multiplied. In order for matrices $\text{[math]}$ and $\text{[math]}$ to be compatible for multiplication, the number of columns in $\text{[math]}$ must equal the number of rows in $\text{[math]}$ . If $\text{[math]}$ is an $\text{[math]}$ matrix and $\text{[math]}$ is an $\text{[math]}$ matrix, the product matrix $\text{[math]}$ will have $\text{[math]}$ rows and $\text{[math]}$ columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for $\text{[math]}$ .

The $\text{[math]}$ identity matrix $\text{[math]}$ satisfies the property:

\text{[math]}

for all $\text{[math]}$ matrices $\text{[math]}$ .

Moreover, every square matrix $\text{[math]}$ with nonzero determinant has an inverse matrix $\text{[math]}$ such that

\text{[math]}

This means that for every positive integer $\text{[math]}$ , the set of all $\text{[math]}$ matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group $\text{[math]}$ .