Inverse Element¶

Definition¶

Let $\star$ be a binary operation on $A$ with identity element $e$ . An inverse element of $a\in A$ is an element $a^{-1}\in A$ such that

a\star a^{-1}=a^{-1}\star a=e.

Interpretation¶

An inverse undoes the effect of an element with respect to the operation: combining $a$ with $a^{-1}$ , in either order, gives the identity element $e$ . The definition therefore only makes sense once an identity exists.

The notation $a^{-1}$ has to be read according to the operation: under addition the inverse of $a$ is $-a$ , under multiplication it is $1/a$ , and for a bijection from a set to itself under composition it is the inverse function. Some elements have none: $0$ has no multiplicative inverse, and in $\mathbb{Z}$ under multiplication only $1$ and $-1$ have one.

When the operation is associative, an inverse is unique: if $b$ and $c$ are both inverses of $a$ , then $b=b\star(a\star c)=(b\star a)\star c=c$ $c$ . Associativity also gives $(a\star b)^{-1}=b^{-1}\star a^{-1}$ $a^{- 1}$ , with the order reversed, as when undoing "socks, then shoes"; for permutations under composition the order matters. Inverses are what turn a monoid into a group.

Key Results¶

A right inverse satisfies $a\star a^{-1}=e$ $e$ .
A left inverse satisfies $a^{-1}\star a=e$ $e$ .

Examples¶

Under addition, the inverse of $a$ is $-a$ . Under multiplication, the inverse of nonzero $a$ is $1/a$ .

Prerequisites¶

Identity Element

Sources¶

Math II: Linear Algebra · Week 1

Keywords: reciprocal element Symbols: element of, integers

Last updated: 20 October 2026 03:02