# Dictionary Definition

invertible adj : having an additive or
multiplicative inverse [ant: non-invertible]

# User Contributed Dictionary

#### Translations

(mathematics) which has an inverse

- Swedish: inverterbar

# Extensive Definition

In mathematics, the idea of
inverse element generalises the concepts of negation,
in relation to addition, and reciprocal,
in relation to multiplication. The
intuition is of an element that can 'undo' the effect of
combination with another given element.

## Formal definition

Let S be a set with a binary
operation *. If e is an identity
element of (S,*) and a*b=e, then a is called a left inverse of
b and b is called a right inverse of a. If an element x is both a
left inverse and a right inverse of y, then x is called a two-sided
inverse, or simply an inverse, of y. An element with a two-sided
inverse in S is called invertible in S. An element with an inverse
element only on one side is left invertible, resp. right
invertible.

Just like (S,*) can have several left identities
or several right identities, it is possible for an element to have
several left inverses or several right inverses (but note that
their definition above uses a two-sided identity e). It can even
have several left inverses and several right inverses.

If the operation * is associative then if an
element has both a left inverse and a right inverse, they are equal
and unique. In this case, the set of (left and right) invertible
elements is a group,
called the group of
units of S, and denoted by U(S) or S^*.

## Calculation

Every real number x
has an additive
inverse (i.e. an inverse with respect to addition) given by -x. Every
nonzero real number x has a multiplicative
inverse (i.e. an inverse with respect to multiplication) given by
\frac 1. By contrast, zero has no
multiplicative inverse.

A function g is the left (resp. right) inverse of
a function f (for function
composition), if and only if g o f (resp. f o g) is the
identity
function on the domain
(resp. codomain) of
f.

A square
matrix M with entries in a field
K is invertible (in the set of all square matrices of the same
size, under matrix
multiplication) if and only if its determinant is different
from zero. If the determinant of M is zero, it is impossible for it
to have a one-sided inverse; therefore a left inverse or right
inverse implies the existence of the other one. See invertible
matrix for more.

More generally, a square matrix over a commutative
ring R is invertible if and
only if its determinant is invertible in R.

Non-square matrices of full rank have
one-sided inverses:

- For A:m\times n \mid m>n we have a left inverse: \underbrace_ A = I_
- For A:m\times n \mid m we have a right inverse: A \underbrace_ = I_

## Example

A:2\times 3 = \begin 1 & 2 & 3 \\ 4 &
5 & 6 \end

So, as mA^_ = A^(AA^)^

AA^ = \begin 1 & 2 & 3 \\ 4 & 5 &
6 \end\cdot \begin 1 & 4\\ 2 & 5\\ 3 & 6 \end = \begin
14 & 32\\ 32 & 77 \end

(AA^)^ = \begin 14 & 32\\ 32 & 77 \end^ =
\frac \begin 77 & -32\\ -32 & 14 \end

A^(AA^)^ = \frac\begin 1 & 4\\ 2 & 5\\ 3
& 6 \end \cdot \begin 77 & -32\\ -32 & 14 \end

= \frac \begin -17 & 8\\ -2 & 2\\ 13
& -4 \end = A^_

The left inverse doesn't exist, because A^A =
\begin 1 & 4\\ 2 & 5\\ 3 & 6 \end \cdot \begin 1 &
2 & 3 \\ 4 & 5 & 6 \end = \begin 17 & 22 & 27
\\ 22 & 29 & 36\\ 27 & 36 & 45 \end

Is a singular matrix, and can't be
inverted.

## See also

## References

invertible in Bulgarian: Обратен елемент

invertible in Czech: Inverzní prvek

invertible in German: Inverses Element

invertible in Spanish: Elemento simétrico

invertible in French: Élément symétrique

invertible in Korean: 역원

invertible in Croatian: Inverzni element

invertible in Italian: Elemento inverso

invertible in Hebrew: איבר הופכי

invertible in Dutch: Inverse element

invertible in Japanese: 逆元

invertible in Polish: Element odwrotny

invertible in Russian: Обратный элемент

invertible in Slovak: Inverzný prvok

invertible in Chinese: 逆元素