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Fearless Symmetry by Robert Gross, Avner Ash

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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
241
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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