Section 26: Associativity and Order of Operations One of the most important properties of the matrix operations is called associativity To understand what this property is we need to discuss something called order of operations In mathematics we are constantly writing mathematical sentences with operations like addition and multiplication Expressions like: 3 + 4 5 (4 7) + 6 6 (4 + 5) Since we have begun to use variables like x in our work, we ll start to use a to symbolize multiplication so there is no confusion All of the operations we are familiar with are called binary operations because they combine 2 objects together As we explained before, binary comes from the Greek word for two But in an expression like 3 + 4 5, what do we do first? Do we add 3 + 4 first or do we multiply 4 5 first? Like so many things in mathematics, the way we do things is simply a tradition that was decided upon many years ago, We have developed the following traditions for the order in which to do the operations: 200 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
1st: Do whatever is inside the parentheses! 2nd: Do multiplication and division next, working from left to right! 3rd: Do addition and subtraction last, working from left to right! Let s look at an example 5 (3+4) - 7 5 7-7 First we do what is in the parentheses Second, we do the multiplication 35-7 Finally, we do the subtraction 28 201 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
This is another, more complicated example Here we have more than one set of parentheses: We went to the parentheses Inside the parentheses, we found more parentheses The most inside parentheses needs to be done first 7 5 + 4 + 3 - (6 + ( 5 4)) Next we finished what what as left in the parentheses 7 5 + 4 + 3 - ( 6 + 20) 7 5 + 4 + 3-26 Next we did the multiplication 35 + 4 +3-26 16 Finally, we did the addition and subtraction, working from left to right 202 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
Name Teacher Date Exercise 261 For each of the expressions below, each team should determine the series of steps needed to perform the computation Each individual should write those steps below, and then teams should be chosen to report to the class on the steps they took, and why they did them in the order they chose (Note: you may not need all the steps listed for a problem) 7 ( 5 + 6 ) + 8 2 1st step 2nd step 3rd step 4th step 5th step (3 + 7 ) (5 ( 7 + 2)) 1st step 2nd step 3rd step 4th step 5th step 203 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
3 ( 6-5 ) + 8 2 + 4 1st step 2nd step 3rd step 4th step 5th step ((3 4) + 8 ) (5 6) 1st step 2nd step 3rd step 4th step 5th step 3 + ( 5 ( 3 + ( 2 4))) 1st step 2nd step 3rd step 4th step 5th step 204 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
Now that we know how to deal with parentheses, let s see if our matrix operations are associative Matrix addition would satisfy the associative property of addition ( ) if anytime I had three matrices, [A], [B], and [C], ([A] + [B]) + [C] = [A] + ([B] + [C]) 205 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
Name Teacher Date Exercise 262 Three matrices, [A], [B], and [C], are given below As a team, decide if these matrices satisfy the associative property for addition Then decide if the associative property is true for any three matrices Report your conjecture to the class, along with some reasons why you think it is true or not 0 7 1 1 1 6 0 8 0 [A] = 2 3 5 [B] = 4 2 4 [C] = 1 2 1 6 3 2 1 1 5 1 3 2 206 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
What about matrix multiplication? Does it also satisfy an associative property? The associative property of multiplication ( ) would be true for matrices if for any three matrices [A], [B], and [C], ([A] * [B]) *[C] = [A] *([B] * [C]) Let s use the road matrices we generated for the multiplication table in section 17 We can experiment with them 207 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
Exercise 263 Name Teacher Date Directions: As a team, decide if matrices from the multiplication table in section 17 atisfy the associative property for multiplication Then form a conjecture about whether the associative property is true for any three road matrices Report your conjecture to the class, along with some reasons why you think it is true If necessary represent the matrices as arrow diagrams and perform the calculations below 208 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
In addition to multiplying road matrices, we defined a technique to multiply other matrices in Section 24 For the challenge problem on the next page, try to decide if this operation also satisfies the associative property of multiplication 209 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc
Exercise 264 Name Teacher Date Directions: Three matrices, [A], [B], and [C], are given below As a team, decide if these matrices satisfy the associative property for multiplication Then decide if the associative property is true for any three matrices Report your conjecture to the class, along with some reasons why you think it is true or not Use the space below for your calculations 0 1 1 0 1 1 [A] = 1 1 [B] = 1 1 [C] = 1 0 210 The Algebra Project Inc Desktop Publishing by, Algebra Project Inc