Section 29: What s an Inverse?
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1 Section 29: What s an Inverse? Our investigations in the last section showed that all of the matrix operations had an identity element. The identity element for addition is, for obvious reasons, called the zero matrix since all of its entries are 0. There is a different size zero matrix for every size matrix, since only matrices of the same size can be added. There are also different size identity matrices for multiplication. We looked at the different representations for these matrices, and one of these was the road subgraph representation. Below is road subgraph of the 4 x 4 identity matrix
2 From our earlier work, we can easily write down the matrix for the road subgraph above. We will start using the symbol I for identity. So below is the 4 x 4 identity matrix [I] = Notice the similarities all of the identity matrices have with one another. The key characteristic in all of them is that there is a diagonal of 1 s going from the top left to the bottom right. These 1 s signify the road loops that we saw in the figure on the previous page. The road loops leave everyone in the same position. 226
3 The identity matrix must always have the same number of rows as columns, just as with the road matrices since each building is both a leave from and go to building. In fact, since matrix multiplication is far more important in mathematics than matrix addition, we call one of these matrices with 1 s along the main diagonal an identity matrix( ). Here is the 5 x 5 identity matrix: [I] 5 x5 = In mathematics, an inverse is always the object that gets us back to the identity element. So to find the inverse of an element we always have to think about certain questions. The first thing we have to understand is what operation we are working with, and the next thing we need to understand is what the identity element is for that operation. 227
4 Let s look at some examples: What is the inverse of an integer when we are doing addition? Question 1: What is the operation? Answer: Addition of integers Question 2: What is the identity element for addition of integers? Answer: The number 0. Remember, the inverse of an integer under addition is the number we add to get back to 0. We use the term additive inverse ( ) for this number. What is the additive inverse of 7? 7 + x = 0 228
5 What is x? The correct value for x is -7 since 7 + (-7) = 0. We say -7 is the additive inverse of 7, since adding -7 to 7 gets us back to the identity element 0. Now let s think about the kind of questions we need to ask to find the multiplicative inverse of a number. 229
6 We always need to understand two things when finding the inverse. What is the operation? Answer: Multiplication of numbers What is the identity element for multiplication of numbers? Answer: The number 1. Suppose we want to find the multiplicative inverse of 20? What number gets 20 back to the number 1 when we multiply? In other words, find x in the equation. 20 x = The correct value for x is, since 20 = We say is the multiplicative inverse of 20, since 20 1 multiplying 20 by gets us back to the 20 identity element
7 What is the multiplicative inverse of 0? To answer this, we try to find an x that satisfies the equation:. 0 x = 1 Is there an answer to this equation? Is there a value for x which makes this equation true?. Since 0 x = 0 for every possible x we can put into this equation, we say that 0 has NO multiplicative inverse. So some numbers don t have inverses! Now what about our matrix operations? Once again the important things to identify are the operation in question and the identity element for the operation. 231
8 To find out if matrices have additive inverses ask the same two questions. What is the operation? Answer: Addition of matrices. What is the identity element for addition of matrices? Answer: The zero matrix. The inverse of a matrix under addition is the matrix we add to get back to the zero matrix. Just as with numbers, we can have matrix equations. To write a matrix equation involving the zero matrix, we will need a symbol for it. Remember, we used the symbol [Z] to refer to the zero matrix. The equation for finding the additive inverse of a matrix [A] would look exactly like the equation for integers [A] + [X] = [Z] Let s have a team exercise to try and find the additive inverse of a matrix 232
9 Name Teacher Date Exercise 29.1 A matrix, [A], is given below. As a team, decide if this matrix has an additive inverse. Also, write an equation that you would have to solve to find the additive inverse of this matrix, [A]. Report to the class whether you found an additive inverse for [A], along with the general equation to find the additive inverse of this matrix [A] =
10 Worksheet: Matrix Additive Inverses Name Teacher Date Exercise 29.2 For each of the matrices below, write the equation that you need to solve to find the additive inverse of the given matrix. Pay special attention to the size of the zero matrix in your equation. Then solve the equation to find the additive inverse. Example : [A] = Equation to solve: [ A] + [X] = Solution: [X] =
11 1) [C] = ) [D] =
12 3) [E] = ) [F] =
13 Let s try to determine what it would mean to find the multiplicative inverse of a road matrix. Remember, we always need to understand two things when finding the inverse. What is the operation? Answer: Multiplication of road matrices What is the identity element for multiplication of road matrices? Answer: The identity matrix The inverse of a road matrix using multiplication, is the matrix we multiply by to get back to the identity matrix. Suppose we started with the road matrix below. Next to it we have drawn its arrow diagram. 237
14 [R] = R To get back to the identity matrix means that we need to find another matrix, say [X], to multiply with [R] so that the product is the identity matrix. In equation form, this would be [R] [X]= [I]. * If we wrote down an arrow diagram equation it would look like this! R X I =
15 Challenge Problem Name Teacher Date Exercise 29.3 The road matrix [R] ifrom the previou page is given below. As a team, try to discover if there is an inverse element for matrix multiplication by finding a matrix [X] that changes [R] into the identity matrix. Write out the arrow diagram equation from the last page and solve to find this inverse matrix, if it exists. Report your conjecture to the class, along with some reasons why you think it is true [R] =
16 Challenge Problem Name Teacher Date Exercise 29.4 The road matrix [R] is given below. As a team, try to discover if there is an inverse element for matrix multiplication by finding a matrix [X] that changes [R] into the identity matrix. Write out the arrow diagram equation and solve it to find this inverse matrix, if it exists. Report your conjecture to the class, along with some reasons why you think it is true [R] =
17 Challenge Problem Name Teacher Date Exercise 29.5 The road matrix [R] is given below. As a team, try to discover if there is an inverse element for matrix multiplication by finding a matrix [X] that changes [R] into the identity matrix. Write out the arrow diagram equation and solve it to find this inverse matrix, if it exists. Report your conjecture to the class, along with some reasons why you think it is true [R] =
18 Name Teacher Date Exercise 29.6 Did your team find a multiplicative inverse for the last matrix? As a team, explain below why or why not you found an inverse matrix.? Can you design other road matrices that will not have an inverse? Report your explanation to the class and report any other road matrices that you have constructed that your team thinks will not have a multiplicative inverse. 242
19 Name Teacher Date Exercise 29.7 For each of the road matrices below, write the equation that you need to solve to find the multiplicative inverse of the given road matrix. Pay special attention to the size of the identity matrix in your equation. Then try to solve the equation to find the multiplicative inverse. If the road matrix has no multiplicative inverse, write no solution. Use arrow diagrams to help you find the inverse. Example : [R] = [R]* [X] = R X I = [X] =
20 1) [R] = ) [R] =
21 ) [R] = ) [R] =
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