434 ALGEBRAIC SYSTEMS [APP. B
(b) Consider the operation of matrix multiplication on the setMofn-square matrices. One can prove that matrix
multiplication is associative. On the other hand, matrix multiplication is not commutative. For example,
[
12
34][
56
0 − 2]
=[
52
15 10]
but[
56
0 − 2][
12
34]
=[
23 34
− 6 − 8]Identity Element:
Consider an operation∗on a setS. An elementeinSis called anidentityelement for∗if, for any element
ainS,
a∗e=e∗a=a
More generally, an elementeis called aleft identityor aright identityaccording ase∗a=aora∗e=awhere
ais any element inS. The following theorem applies.
Theorem B.2: Supposeeis a left identity andfis a right identity for an operation on a setS. Thene=f
The proof is very simple. Sinceeis a left identity,ef =f; butsincefis a right identity,ef =e. Thus
e=f. This theorem tells us, in particular, that an identity element is unique, and that if an operation has more
than one left identity then it has no right identity, and vice versa.
Inverses:
Suppose an operation∗on a setSdoes have an identity elemente. Theinverseof an elementainSis an
elementbsuch that
a∗b=b∗a=e
If the operation is associative, then the inverse ofa, if it exists, is unique (Problem B.2). Observe that ifbis the
inverse ofa, thenais the inverse ofb. Thus the inverse is a symmetric relation, and we can say that the elements
aandbare inverses.
Notation:If the operation onSis denoted bya∗b,a×b,a·b, or ab, thenSis said to be writtenmultiplicatively
and the inverse of an elementa∈Sis usually denoted bya−^1. Sometimes, whenSis commutative, the operation
is denoted by+and thenSis said to be writtenadditively. In such a case, the identity element is usually denoted
by 0 and it is called thezeroelement; and the inverse is denoted by−aand it is called thenegativeofa.
EXAMPLE B.4 Consider the rational numbersQ. Under addition, 0 is the identity element, and−3 and 3 are
(additive) inverses since
(− 3 )+ 3 = 3 +(− 3 )= 0
On the other hand, under multiplication, 1 is the identity element, and−3 and− 1 /3 are (multiplicative) inverses
since
(− 3 )(− 1 / 3 )=(− 1 / 3 )(− 3 )= 1
Note 0 has no multiplicative inverse.
Cancellation Laws:
An operation∗on a setSis said to satisfy theleft cancellation lawor theright cancellation lawaccord-
ing as:
a∗b=a∗cimpliesb=c or b∗a=c∗aimpliesb=c
Additionandsubtraction of integers inZand multiplication of nonzero integers inZdo satisfy both the left
and right cancellation laws. On the other hand, matrix multiplication does not satisfy the cancellation laws. For
example, suppose
A=[
11
00]
,B=[
11
01]
,C=[
0 − 3
15]
,D=[
12
00]ThenAB=AC=D, butB=C.