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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the idea of inverse element generalises the concepts of negation
Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
, in relation to addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, and reciprocal
Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
, in relation to multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
. The intuition is of an element that can 'undo' the effect of combination with another given element.

Formal definition
Let be a set with a binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 . If is an identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 of and , then is called a left inverse of and is called a right inverse of . If an element is both a left inverse and a right inverse of , then is called a two-sided inverse, or simply an inverse, of .






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the idea of inverse element generalises the concepts of negation
Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
, in relation to addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, and reciprocal
Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
, in relation to multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
. The intuition is of an element that can 'undo' the effect of combination with another given element.

Formal definition


Let be a set with a binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 . If is an identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 of and , then is called a left inverse of and is called a right inverse of . If an element is both a left inverse and a right inverse of , then is called a two-sided inverse, or simply an inverse, of . An element with a two-sided inverse in is called invertible in . An element with an inverse element only on one side is left invertible, resp. right invertible.

Just like 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 ). 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
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, called the group of units of , and denoted by or .

A left-invertible element is left-cancellative
Cancellation property

In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c....
, and analogously for right and two-sided.

Calculation


Every real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
  has an additive inverse
Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
 (i.e. an inverse with respect to addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
) given by . Every nonzero real number has a multiplicative inverse
Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
 (i.e. an inverse with respect to multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
) given by . By contrast, zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
 has no multiplicative inverse.

A function is the left (resp. right) inverse of a function (for function composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
), if and only if (resp. ) is the identity function
Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
 on the domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 (resp. codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
) of .

A square matrix with entries in a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
  is invertible (in the set of all square matrices of the same size, under matrix multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
) if and only if its determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 is different from zero. If the determinant of 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
Invertible matrix

In linear algebra, an n-by-n matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that...
 for more.

More generally, a square matrix over a commutative ring
Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  is invertible if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 its determinant is invertible in .

Non-square matrices of full rank have one-sided inverses:
  • For we have a left inverse:
  • For we have a right inverse:
No rank-deficient matrix has any (even one-sided) inverse. However, the Moore-Penrose pseudoinverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists.

Example



So, as m

The left inverse doesn't exist, because , which is a singular matrix, and can't be inverted.

See also

  • loop
  • division ring
    Division ring

    In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. More formally, a ring with 0 ? 1 is a division ring if every non-zero element a has a multiplicative inverse ....
  • unit (ring theory)
    Unit (ring theory)

    In mathematics, a unit in a ring R is an invertible element of R, i.e. an element u such that there is a v in R withThat is, u is an invertible element of the multiplicative monoid of R....