MATRICES 117

We call the actual numbers that occur in a given matrix its

entries. So the last exchange can be stated: “What was the entry

in the second row and ﬁrst column? Oh, it was a 0?” Or when you

get good at it, you can even say “What was the (2,1)-entry?”

Matrix Multiplication

Because the entries in a matrix are elements of a number system,

they can be added, subtracted, and multiplied. This allows us to

deﬁne multiplication of matrices, and thus we will obtain plenty of

“standard” groups to use in our representations.

Here is the all-important deﬁnition of how to multiply two

matrices. First, we have to tell you that any two matrices cannot

necessarily be multiplied. They must both have entries from the

same number system. For instance, both could be Z-matrices, or

Q-matrices, or C-matrices.

The sizes of the matrices also have to match in a particular way.

Remember that an m-by-n matrix is a rectangular array of numbers

with m rows and n columns. Let A be an m-by-nR-matrix and

B a p-by-qR-matrix. Then we can form and compute the product

AB (in that order!) if and only if n = p. In words: The number

of columns of the ﬁrst matrix must be the same as the number of

rows of the second matrix. You will see why as soon as we explain

how to multiply.

1

A mnemonic for this rule is that the numbers

in the middle have to “cancel out.” In fact, an m-by-n matrix

times an n-by-q matrix will be an m-by-q matrix, the n’s having

cancelled out.

It will be helpful ﬁrst to describe the dot product of a row and

column of the same sizes. We assume the number system is ﬁxed

for this discussion, so that we can add and multiply entries. The dot

product of a row of length n and a column of length n is obtained

by multiplying the pairs of entries in corresponding positions in

1

You may have noticed that we do not use any symbol for the group law of multiplying

matrices. We just juxtapose the symbols of the two matrices, as in AB. This notation is

traditional.

118 CHAPTER 10

the row and column, and then adding up all of those answers. For

example:

The dot product of

135

and

⎡

⎢

⎣

2

9

1

⎤

⎥

⎦

is 1 · 2 + 3 · 9 + 5 · 1 =

2 + 27 + 5 = 34.

So the dot product of a row and a column (of the same size) is a

single number.

Next, let’s multiply a 2-by-2 Z-matrix A by a 2-by-1 Z-matrix B:

25

41

6

8

=

2 · 6 + 5 · 8

4 · 6 + 1 · 8

=

52

32

.

The answer is another 2-by-1 Z-matrix. The multiplication rule

can be stated in words as follows: The ﬁrst entry in the answer is

the dot product of the ﬁrst row of A by B, and the second entry in

the answer is the dot product of the second row of A by B.

Or here’s a 4-by-3 Z-matrix times a 3-by-1 Z-matrix:

⎡

⎢

⎢

⎢

⎣

123

781

−123

1110

⎤

⎥

⎥

⎥

⎦

⎡

⎢

⎣

40

28

31

⎤

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

1 · 40 + 2 · 28 + 3 · 31

7 · 40 + 8 · 28 + 1 · 31

−1 · 40 + 2 · 28 + 3 · 31

1 · 40 + 11 · 28 + 0 · 31

⎤

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

189

535

109

348

⎤

⎥

⎥

⎥

⎦

.

Again, each entry in the answer is a dot product.

In general, an m-by-n matrix times an n-by-1 matrix will be an

m-by-1 matrix computed according to this rule: the kth entry of the

answer is the dot product of the kth row of the ﬁrst matrix times

the column matrix.

EXERCISE: Compute the following products:

⎡

⎢

⎣

56

34

11 −2

⎤

⎥

⎦

4

9

434

182

⎡

⎢

⎣

7

9

12

⎤

⎥

⎦

.

MATRICES 119

SOLUTION: The answers are

⎡

⎢

⎣

56

34

11 −2

⎤

⎥

⎦

4

9

=

⎡

⎢

⎣

74

48

26

⎤

⎥

⎦

and

434

182

⎡

⎢

⎣

7

9

12

⎤

⎥

⎦

=

103

103

.

Now multiplying a matrix by another matrix (of the right size) is

easy. An m-by-n matrix times an n-by-q matrix will be an m-by-q

matrix computed according to this rule: the (k, j)th entry of the

answer is the dot product of the kth row of the ﬁrst matrix and

the jth column of the second matrix.

That was a mouthful, so let us take a look. We have

12

34

56

=

1 · 3 + 2 · 51· 4 + 2 · 6

=

13 16

.

Here is another example:

3411

24−1

⎡

⎣

45 6 11

22 1 34

943237

⎤

⎦

=

3 ·4 + 4 ·2 + 11 ·93·5 + 4 ·2 + 11 ·43·6 + 4 ·1 + 11 ·32 3 ·11 + 4 ·34 + 11 ·37

2 ·4 + 4 ·2 +−1 ·92·5 + 4 ·2 +−1 ·42·6 + 4 ·1 +−1 ·32 2 ·11 + 4 ·34 +−1 ·37

=

119 67 374 576

714

−16 121

.

EXERCISE: Compute the following matrix products:

7811

4312

⎡

⎢

⎣

85

39

11 2

⎤

⎥

⎦

⎡

⎢

⎣

85

39

11 2

⎤

⎥

⎦

7811

4312

.

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