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 A’s 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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