ME/CS 133(a): The Classical Matrix Groups

Note that the above definition introduces the identity e as a left identity (i.e., the identity multiplies a group element on the left). Similarly, the inverse of group element gi was defined as a left inverse, where the inverse element multiplies the group element on the left. The group definition can be used to show that e is also a right identity (i.e., e ∗ g = g ∗ e = g) and g−1 is a right inverse of g (g−1 ∗ g = g ∗ g−1 = e). Proof: (that the left inverse g−1 is also a right inverse) g−1 = e ∗ g−1 = (g−1 ∗ g) ∗ g−1 = g−1 ∗ (g ∗ g−1) (1)