The Road Coloring Problem
Section 27: Commutative Property
If an operation satisfies the associative property, then it doesn’t
matter in which time-order we do things. If we are given 3 + 4 + 5,
we could first do the 3 + 4 to get 7 , and second add the 5 to get 12.
We could also first do the 4 + 5 to get 9, and second add 3 + 9 to get 12.
But notice that one thing didn’t change: the position of the numbers as we
read the expression from left to right. We changed the
time we did things, but not the position.
An operation satisfies the commutative property,
if we can switch the position of the terms in
an expression. For example, we know that the set
of integers satisfies the commutative
property of addition since, for example:
2 + 6 = 6 + 2.
The integers also satisfy
the commutative
property of multiplication
since,for example:
3
. .
5=5
3.
211
The Algebra Project Inc. © Desktop Publishing by, Algebra Project Inc. ©
The Road Coloring Problem
What about our
matrix operations?
Matrix addition would satisfy the commutative
property of addition (
) if anytime
we had two matrices, [A] and [B]
[A] + [B] = [B] + [A]
The road matrices would satisfy the commutative
property of multiplication (
) if
anytime we had two matrices, [A] and [B]
[A] * [B] = [B] * [A]
212
The Algebra Project Inc. © Desktop Publishing by, Algebra Project Inc. ©
The Road Coloring Problem
Name __________________ Teacher __________________ Date _____
Exercise 27.1
Two matrices [A] and [B] are given below. As a team, decide if these matrices satisfy
the commutative property for addition. Then decide if the commutative property for
addition is true for any two matrices. Report your conjecture to the class, along with
some reasons why you think it is true or not.
1
[A] =
1
–
2
6
0
5
7
3
4
[B] =
1
2
5
6
4
3
9
0
4
–
213
The Algebra Project Inc. © Desktop Publishing by, Algebra Project Inc. ©
The Road Coloring Problem
Name __________________ Teacher __________________ Date _____
Exercise 27.2
Two road matrices [A] and [B] are given below. As a team, decide
if these matrices satisfy the commutative property for multiplication. Then decide
if the commutative property is true for any two road matrices. Report your
conjecture to the class, along with some reasons why you think it is true or not.
0
[A] =
1
0
1
0
0
0
0
1
[B] =
1
0
0
0
0
1
0
1
0
214
The Algebra Project Inc. © Desktop Publishing by, Algebra Project Inc. ©