Matrix

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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.

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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 = 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}).

Matrix concepts

Basic concepts

Advanced concepts

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