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Fearless Symmetry by Robert Gross, Avner Ash

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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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