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Fearless Symmetry by Robert Gross, Avner Ash

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126 CHAPTER 11
multiplication in this set: It is called the identity matrix.Ithas
1’s down the main diagonal that goes from upper left to lower right
(this is simply called the diagonal” when dealing with matrices)
and it has 0’s everywhere else. For instance, here is the 3-by-3
identity matrix:
100
010
001
.
It is traditional to call this matrix I. Of course, if you change n, you
change I, so it should really be called I
n
, but the size of I will be
clear from the context.
EXERCISE: Take this 3-by-3 matrix I and multiply it by any
3-by-3 matrix A. Try it on the left and the right: AI and IA
both will turn out to be A again.
Matrix Inverses
Great! All we need are inverses. But now things get a little sticky,
or, as a mathematician would say, interesting. Not every matrix has
an inverse.
DEFINITION: An n-by-n matrix A is invertible,orhas an
inverse, if there is some n-by-n matrix H with the property
that AH and HA both equal I, the n-by-n identity matrix. If
this is so, we say that H is the inverse of A.
Here is a 3-by-3 example of an invertible matrix A:
234
102
348
.
GROUPS OF MATRICES 127
And here is its inverse H:
4
3
4
3
1
1
3
2
3
0
2
3
1
6
1
2
.
EXERCISE: Check that AH = HA = I.
It is a fact, easy to prove, that if A is invertible, then it has one
and only one inverse.
3
This we call the inverse of A, and we write it
A
1
. Notice this doesn’t mean
1
A
, for A is not a number, so it cannot
be divided into the number 1.
We claimed that not every square matrix has an inverse. Here is
a stupid example. If C is the 3-by-3 matrix of all 0’s, then CB = C
no matter what 3-by-3 matrix B is. So CB could never equal I. But
there are less stupid examples. If you think about it a little, you
can see that even if just one row of A is all 0’s, then A has no
inverse matrix. Or if two of the rows of A are identical, then A has
no inverse matrix. Or if two of the columns of A are identical, then A
has no inverse matrix. Do you see why?
4
The general theory of which As are invertible and which are not
is a very interesting part of matrix theory. The complete answer is
too much off the track for us to go into it here, but if you happen to
know how to compute the determinant of a square matrix, then you
can understand the following:
THEOREM 11.1: If A is a square R-matrix, then A is
invertible if and only its determinant (which is an element
of R) has a multiplicative inverse in R. That is to say, if d is
the determinant of A, then A is invertible if and only if there
is some element c in R so that the product cd = 1.
3
PROOF:IfH and K are both inverses of A, then H = HI = H(AK) = (HA)K = IK = K.
4
For example, if A has two identical rows, then so does AB for any matrix B. But the
identity matrix does not have two identical rows. So AB can never equal I, whatever
matrix B you try.

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