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Fearless Symmetry by Robert Gross, Avner Ash

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120 CHAPTER 10
SOLUTION: We have
7811
4312
85
39
11 2
=
201 129
173 71
and
85
39
11 2
7811
4312
=
76 79 148
57 51 141
85 94 145
.
The process by which we have defined matrix multiplication is
typically mathematical. First we define and understand a special
case: a row times a column, that is, a 1-by-n matrix times an n-by-1
matrix, will be a 1-by-1 matrix—a single number—given by the dot
product. From this we build up: next defining any matrix times a
column, and finally any matrix times any matrix (as long as the
sizes match).
Linear Algebra
One simple use of matrices is to solve systems of Z-equations
where none of the exponents is higher than 1. This is called “linear
algebra.
2
Here is an example.
Suppose we want to solve the system of Z-equations:
3x 5y = 2. (10.1)
2x + 3y = 14. (10.2)
This is easy to solve using elementary algebra. And it is also easy
to graph the two lines (which is why it is called linear algebra) and
see where they intersect. But in order to illustrate matrices, let’s
set this up as a matrix problem.
2
This is one reason why “linear” is often used as an adjective in place of “matricial.
MATRICES 121
We define the Z-matrix A to be
A =
3 5
23
and we define another Z-matrix B to be
B =
2
14
.
Finally, we define an unknown matrix Z in terms of x and y:
Z =
x
y
.
The value Z is just as unknown as the x and y we are trying to find.
Now by practicing your matrix multiplication, you can see that
the system given by (10.1) and (10.2) above is completely equivalent
to the single matrix equation:
AZ = B.
It turns out there is a way to divide by the matrix A and solve
for Z. We will discuss this in more detail in the next chapter. For
now, we just define two more matrices:
A
1
=
3
19
5
19
2
19
3
19
and
I =
10
01
.
Check that A
1
A = I and IZ = Z.
If AZ = B then A
1
(AZ) = A
1
B. In the next chapter we will find
out that matrix multiplication is associative, so that A
1
(AZ) =
(A
1
A)Z = IZ = Z. We conclude that if AZ = B,then
Z = A
1
B =
3
19
5
19
2
19
3
19

2
14
=
76
19
38
19
=
4
2
.

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