We continue to study matrices. Some of them are

“invertible,” which means they can belong to multiplica-

tive groups of matrices. These matrix groups are the

standard objects we referred in the road map of the

previous chapter.

In particular, for any number system R, we deﬁne the

group GL(n, R) of all invertible R-matrices with n rows

and n columns. As an example, we explore GL(2, Z), an

inﬁnite group of special importance in number theory.

Square Matrices

From now on, we will assume that the set of entries in our matrices

consists of elements of a number system, which we will call R. This

means that we can add, subtract, and multiply the elements of R,

and they obey the usual laws of arithmetic. We will not need to use

division at all, and so letting R = Z, for example, is ﬁne.

The entries need not be actual numbers of the ordinary sort.

For example, we could use the ﬁeld F

2

consisting of 0 and 1. Let

us review a part of chapter 4. We deﬁne addition as usual, and

multiplication as usual, with one important exception: 1 + 1 = 0.

(Because there is no “2” in F

2

, we have to redeﬁne 1 + 1. When

someone tells you that a given fact is as obvious as 1 + 1 = 2, tell

her that maybe she should go visit F

2

.) You can ﬁgure out how

GROUPS OF MATRICES 125

to redeﬁne subtraction also.

1

Then all the laws of arithmetic hold:

associative laws of addition and multiplication, commutative laws

of addition and multiplication, distributive laws, adding 0 does not

change the number you add it to, and neither does multiplying

by 1. Later in this chapter, we will study the group GL(2, F

2

).

We explore the following question: When is a set of R-matrices a

group, with the group law given by matrix multiplication? Remem-

ber that this means (see chapter 2):

1. The set is “closed” under the group operation.

2

2. The associative law holds.

3. There is a neutral element.

4. Any element in the set has an inverse element in the set.

The ﬁrst thing to verify is the nonobvious fact that matrix

multiplication is always associative. You can ﬁnd the proof in any

linear algebra textbook. Even if you do not want to prove it, you can

check an example or two for yourself. Pick three Z-matrices, A, B,

and C of matching sizes: the number of columns of A = the number

of rows of B, and the number of columns of B = the number of rows

of C. Then compute (AB)C—multiply AB ﬁrst and then multiply the

result by C on the right. Next compute A(BC)—multiply BC ﬁrst

and then multiply the result by A on the left. If you did not make

any errors, the answers should be the same.

By now, you may have noticed that the set of all R-matrices

is not closed under matrix multiplication for the unusual reason

that you cannot even multiply two matrices if their sizes do not

match. A group law has to be able to take any two elements of

the group and combine them to get a third element of the group.

If we try to limit our set of matrices to some set where every matrix

matches every other one in the set for multiplicative purposes, we

see that we have to limit ourselves to square matrices of some ﬁxed

size.

So ﬁx a positive integer n, and now let us consider only n-

by-nR-matrices. There is indeed a neutral element for matrix

1

The only deviation from usual subtraction is that 0 − 1 = 1.

2

DEFINITION:AsetS is closed under a group law if x ◦ y is in S whenever both x and y

are in S.

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