130 CHAPTER 11

DEFINITION: GL(n, R) is the group of all n-by-n invertible

R-matrices, where the group law is given by matrix

multiplication.

For example, the six matrices (11.2) form the group GL(2, F

2

).

Another thing to remember about matrix multiplication is that

it usually is not commutative. For example, in GL(2, Q)wehave

12

34

56

78

=

19 22

43 50

and

56

78

12

34

=

23 34

31 46

.

Another example drawn from GL(2, F

2

):

11

01

01

10

=

11

10

and

01

10

11

01

=

01

11

.

The Group GL(2 , Z)

The group GL(2, Z) is one way in which number theory and group

theory can come together, though we can give only a few details in

this book. For example, the theory of modular forms, referred to in

chapters 21–23, depends crucially on GL(2, Z) and its properties.

In this section, we will see that even the simple problem of enu-

merating the elements of GL(2, Z) depends on elementary number

theory.

We know that GL(2, Z) is the set of 2-by-2 matrices with integer

entries and determinant equal to 1 or −1. (This is because 1 and −1

are the only elements in Z that have multiplicative inverses in Z.)

It is not that hard to ﬁnd elements of GL(2, Z), after you know that

the determinant of A =

ab

cd

is ad − bc. The entries a, b, c, and

d must all be integers, and the only requirement for invertibility

is that ad − bc = 1orad − bc =−1. We investigate the case where

ad − bc = 1. Then A

−1

is given by

d −b

−ca

. For example, we have

35

47

−1

=

7 −5

−43

.

GROUPS OF MATRICES 131

How do we construct matrices in GL(2, Z)? We start by picking

the ﬁrst row at random: say [6 15]. Then we look for the second

row [cd] to satisfy 6d − 15c = 1. Whoops. That can never happen

if c and d are integers, because the left-hand side is an exact

multiple of 3, and so it could never equal 1. We had better make sure

that a and b share no common factor. Two numbers like that are

called relatively prime. For example, 3 and 20 are relatively prime,

100 and 57 are relatively prime, but 37 and 111 are not relatively

prime, because both are multiples of 37. (Yes, a number is always

considered to be a multiple of itself.)

Armed with this knowledge, we now pick our ﬁrst row so the two

entries in it are relatively prime: say [6 11]. Now we look for [cd]

to satisfy 6d − 11c = 1. One way to do this is to list a lot of multiples

of 6 and 11 and look for entries in the list that are only one

apart:

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, ...

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, ...

We immediately notice the 11 on the bottom row and the 12

on the top row, which leads us to choose c = 1 and d = 2. Check:

6 × 2 − 11 × 1 = 1. We also see 78 on the top list and 77 on the

bottom, leading to the choice of c = 7 and d = 13. Check: 6 × 13 −

11 × 7 = 78 − 77 = 1. If we are clever, we can even exploit the 55

on the bottom row and 54 on the top row. The problem is that

54 − 55 =−1, not 1. But we can solve this by multiplying through

by −1: in fact −54 − (−55) = 1. This leads to the choice c =−5 and

d =−

9. Check: 6 × (−9) − 11 × (−5) =−54 − (−55) = 1. We thus

get the following matrices in GL(2, Z):

611

12

,

611

713

,

611

−5 −9

.

We see that not only have we succeeded, but we have also found

several solutions to our problem. This illustrates a general theorem:

If a and b are relatively prime integers, then there are inﬁnitely

many pairs of integers c and d with the property ad − bc = 1.

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