Writing $1/3$ is just a shorthand notation for $1 \cdot 3^{-1}$ in the rational (or real or complex) group; of course you can define it in any other group (like the one you have provided) but using it can be confusing. The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c). In fact, if a is the inverse of b, then it must be that b is the inverse of a. Inverses are unique. Example 3.12 Consider the operation ∗ on the set of integers deﬁned by a ∗ b = a + b − 1. The inverse of ais usually denoted a−1, but it depend on the context | for example, if we use the symbol ’+’ as group operator, then −ais used to denote the inverse of a. Inverse element For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. The identity matrix for is because . Every element has an inverse: for every member a of S, there exists a member a −1 such that a ∗ a −1 and a −1 ∗ a are both identical to the identity element. Similarly, an element v is a left identity element if v * a = a for all a E A. Make a note that while there exists only one identity for every single element in the group, each element in the group has a different inverse. For example, the operation o on m defined by a o b = a(a2 - 1) + b has three left identity elements 0, 1 and -1, but there exists no right identity element. The Identity Matrix This video introduces the identity matrix and illustrates the properties of the identity matrix. Actually that is what you are looking for to be satisfied when trying to find the inverse of an element, nothing else. The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. ... inverse or simply an inverse element. One can show that the identity element is unique, and that every element ahas a unique inverse. A n × n square matrix with a main diagonal of 1's and all other elements 0's is called the identity matrix I n. If A is a m × n matrix, thenI m A = A and AI n = A. It is called the identity element of the group. Back in multiplication, you know that 1 is the identity element for multiplication. Thus, there can only be one element in Rsatisfying the requirements for the multiplicative identity of the ring R. Problem 16.13, part (b) Suppose that Ris a ring with unity and that a2Ris a unit More explicitly, let S S S be a set and ∗ * ∗ be a binary operation on S. S. S. Then This is also true in matrices. -5 + 5 = 0, so the inverse of -5 is 5. Identity element There exists an element e in G such that, for every a in G, one has e ⋅ a = a and a ⋅ e = a. Whenever a set has an identity element with respect to a binary operation on the set, it is then in order to raise the question of inverses. Is A is a n × n square matrix, then Whenever the identity element for an operation is the answer to a problem, then the two items operated on to get that answer are inverses of each other.. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity, i.e., in a semigroup.. You can't name any other number x, such that 5 + x = 0 besides -5. An identity element in a set is an element that is special with respect to a binary operation on the set: when an identity element is paired with any element via the operation, it returns that element. Such an element is unique (see below).