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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. ©