Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices

Size: px
Start display at page:

Download "Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices"

Transcription

1 Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7 Inverse matrices What you need to know already: How to add and multiply matrices. What elementary matrices are. What you can learn here: What the multiplicative inverse of a matrix is and how to construct it. A few, very basic properties of the inverse of a matrix. Now that you know how to add, subtract and multiply matrices, are you curious about how to divide them? I would be, but I am afraid that it will be very complicated, given how messy it is to multiply them! You are right: the matrix product is such a complex operation that devising an effective way to divide matrices has proven to be a futile task in general. Good! Let s move on to the next section Not so fast! It turns out that, while in general it is a futile task, in some not-sorare cases it can be done, but the trick is to think of division in a way that is familiar to you already. The key is to remember that in usual algebra, dividing by a number is the same as multiplying by its reciprocal: b b a a So, instead of constructing some complicated division process, we ll try to give meaning to the concept of reciprocal. We shall do so by focussing on the key property that the product of a number and its reciprocal is : a a a a. Therefore, we shall start by defining the inverse A of a matrix A as being a matrix with this same property. Later we ll try to figure out when and how such inverse matrix can be computed. Definition The inverse of a matrix A is a matrix denoted by A such that: A A AA I A matrix that has an inverse is called an invertible matrix. Does that mean that not all matrices have an inverse? Yes, and that should not be a surprise. After all, not every number has a reciprocal: 0 does not! This is a major issue that we shall explore at length later. But we begin by looking some other issues that must be noted right away. We begin with the minor one. Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page

2 Knot on your finger Although the matrix A has been defined to mirror the properties of a reciprocal, it is called an inverse instead. The reason for this choice will become clear once we use inverse matrices in a role for which they represent inverse functions in the calculus meaning of the word. Part of the fun of mathematics is to determine how little you need to assume in order to make something work. Some mathematicians spend their whole research career on this kind of problems! But let us notice a more pressing issue for us. Knot on your finger Since matrix multiplication is not commutative, even if we find a matrix A for which A A I, we must still check that AA I. And now for two more technical issues. In order for a matrix A to be invertible, it must be a square matrix. And that will make our search more complicated, right? Not necessarily, but it will require us to check this point. I get the problem with the commutativity, but how do we even find an inverse? It still seems very messy to reverse a matrix product. It seems so, but it turns out that if we think carefully about the properties of the matrix product that we have seen so far, developing a suitable method will be pretty simple and the method will not be complicated on unfamiliar either! So, let s head in that direction by identifying a property that any invertible matrix must satisfy. Proof If A is an nm matrix, in order for both A A and AA to be defined, A must be an mn matrix. But in that case the products will have dimensions mm and nn respectively. For these to be the same identity matrix, m must be equal to n and hence A must be square. I notice now that in the definition you did not specify which identity matrix needed to be used, but you did specify that it had to be the same one! Sneaky! If a matrix A is invertible, then its RREF must be the identity matrix I. Equivalently, if the RREF of a square matrix A is not the identity, the matrix is not invertible. Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 2

3 Proof We prove this by contradiction. So let us assume that the RREF of A is not I. This means that the homogeneous system Ax=0 has a free variable and hence non trivial solutions. Let us pick one non-zero solution and call it c, so that Ac=0. If, by contradiction, A has an inverse, we must have that: 0 A 0 A Ac Ic c But we picked c as a non-zero vector, so this cannot also be 0! The contradiction implies that A cannot possibly have an inverse. So, in order to look for invertible matrices, we must look among those square matrices whose RREF is I. Can we say that a square matrix whose RREF is I is invertible? Not yet! Notice that this only makes a statement in one direction: if A is invertible, its RREF must be I, but it makes no claim in the opposite direction. It turns out that the opposite direction is indeed true, but we have not proved that yet. To prove it and to construct inverses we will use the elementary matrices we explored in the previous section. To begin, let me show you that elementary matrices are invertible. Proof A Every elementary matrix E has an inverse E. This inverse E is also an elementary matrix and it corresponds to the ERO that reverses the ERO corresponding to E. There are three types of elementary matrix, corresponding to the three types of ERO s. If E is obtained by switching two rows, applying it twice will revert the rows back to their original position. In other words, applying it twice will produce no change, hence produces the identity matrix, which is what we want. So, in this case E E. If E is obtained by multiplying a row by the non-zero scalar k, consider the elementary matrix that multiplies that same row by /k. Again, applying these two matrices one after the other will revert that row back to its original values. In other words, their product corresponds to the identity matrix, which is what we want. Alternatively, just multiply two such matrices together (in the general case!) to see that it works. Finally, if E is obtained by adding to the i-th row a multiple of the j-th row, consider the elementary matrix that subtracts from the i-th row the same multiple of the j-th row. Again, applying these two matrices one after the other will revert the i-th row back to its original values. In other words, their product corresponds to the identity matrix, which is what we want. So, in all three cases the inverse exists and is the claimed elementary matrix. I think I follow the logic, but I still don t see how it works. I am not surprised, since the above proof is rather theoretical. So, if you trust that proof, here is what this fact is saying for 33 elementary matrices. Quick portrait of Inverses of elementary matrices For elementary matrices that switch rows: E E E Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 3

4 For elementary matrices that multiply a row by a non-zero scalar k: E E 0 0 k 0 0 k For elementary matrices that add to a row a multiple of another row: E 0 k 0 k E I have a suggestion for you here: answer the Learning questions that asks you to verify that this portrait is correct. It is a very simple task and will give you a better insight into inverse matrices and how they work in this simple situation, in addition of convincing you that this portrait is accurate. Done? Good. It turns out that this simple fact about elementary matrices allows us to construct the inverse of any matrix whose RREF is I, thus showing also that any such matrix is invertible. The construction that I will show you is based on the following facts that we have learned so far about matrices. To go from a matrix A to its RREF we need to apply a sequence of ERO s. Regardless of which sequence we use, the RREF of A is unique. Each ERO can be implemented by multiplying the current matrix, on the left, by the elementary matrix corresponding to that ERO. The matrix we are looking for must have the property that when multiplied on the left of A, the resulting product is the identity matrix. We are now ready for the main event of this section. Proof If the RREF of a square matrix A is the identity matrix, and if E, E2,..., E p is a sequence of elementary matrices that changes A to I, then: A E pe p E2E is a matrix that works as an inverse for A on the left side. We need to show that A A I, meaning that p p 2 E E E E A I. But this follows from the fact that these elementary matrices were chosen exactly to reduce A to its RREF and that such RREF is I. Notice how, despite our initial fears, this construction is very simple. All we need to do to get the inverse of A is multiply together all the elementary matrices that are needed to reduce A to I, in the proper order! But, according to what you said earlier, we cannot call this an inverse yet, sine it only works when multiplied on the left side. What about the right? I am glad you noticed and we do need to check what happens on the right. However, I will show you that the matrix constructed in this way works on the right side as well. Therefore, it is the inverse and we can identify it as A. OK, but then I beg to disagree about the simplicity of the procedure! You want me to get the RREF, write down all the corresponding matrices and then multiply them together?! That is not very simple! Not in the way you are suggesting, but there is a trick that allows us to do the whole procedure without ever writing down an elementary matrix! Watch: Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 4

5 Strategy for Computing the inverse of a matrix Given a square matrix A: Augment it with the identity of the same dimension: A I Select an appropriate sequence of ERO s in order to obtain the RREF of A and apply each of them to the whole augmented matrix. If the RREF of A is not I, we know that A is not invertible and we can stop. If the procedure generates I on the left side of the augmented matrix, the matrix on its right side is A, that is, the final augmented matrix so obtained is of the form: RREF A I I A. Slow down! This feels like magic. How do I know that the right side is the inverse? Good point. Think about it: by performing all the needed ERO s, we have multiplied the augmented matrix by the corresponding sequence of elementary matrices, that is, EpEp E E 2. On the left side we multiplied this product of matrices by A, so we got I. On the right side we multiplied that product by I, so we got the product back again. But we have seen that this product must be the inverse of A! Bingo! And that does the trick. Here is an example. Example: 2 2 A Let s see if we can find the inverse of this matrix, or if it turns out that it is not invertible. We start by augmenting it with the identity: A I Now we perform a sequence of row operations aiming to get an RREF on the left side, but applying them to the right side as well: r r 2 A I r2 2r r2 r3 r3 4r r r r r r r r 3r2 r2 r r r / / 3 2 r 3 ; ; / / / /3 Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 5

6 And there you have it: check, by multiplication, that in fact the matrix 7 / / 3 8 / / 3 A 5 / / 3 is worthy to be considered as the inverse of our original matrix A. I see that it is not as bad as it seemed, we just have to compute the RREF of the augmented matrix. Yes, and remember that once we know that this is what must be done, we can get assistance from a calculator or computer. Yeeah for computers! And the smart people who program them, whose work we can better appreciate by knowing what is behind it. Now, before we close this section, what do we need to do? Have a chocolate bar? Good idea, but there is still a technical issue to resolve: you haven t forgotten that we still need to check that this matrix works on the right every time, have you? To check that it does we begin by observing a simple, but important property of inverses. Proof If 2 A, A,..., A p are invertible matrices of the same dimension, then so is their product: A A A A 2 p Moreover, the inverse of this product is: A A A A A p p 2 All we need to do is check. If we multiply this product on the left of A, we have: A A A A A A A p p 2 2 p A A A A A A A A p p 2 2 p p A A A IA A A p p 2 2 p p A A I A A A IA I p p p p p p The same works on the right, as I am sure you will be glad to check. But we are still multiplying on the left! Patience, we are not there yet. We have seen that elementary matrices are invertible; therefore, we can apply this last fact applies to a product of them. This allows us to check that the matrix we construct as the inverse, is, in fact, so on both sides. Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 6

7 If A E pe p E2E is the product of elementary matrices obtained in the strategy for constructing the inverse of a matrix A, then: A E E E E A A 2 p p I Therefore, this matrix is indeed the inverse of A as it satisfies the defining property on both sides. Wow! I will have to read this again before I am convinced that it is not magic! You should certainly do that. And to conclude this fairly technical section, here are some useful properties of inverses whose easy proofs are left for you in the Learning questions. After all, a technical section deserves technical Learning questions, no? A square matrix is invertible if and only if its RREF is I. Proof We know that EpEp E2E is a product of elementary matrices and all elementary matrices are invertible. Therefore, this is a product in invertible matrices. From what we have seen before, the product we have constructed is invertible and we know that its inverse is E E E E. But this 2 p product, when applied to I, undoes, in the correct order, all the ERO s we used to go from A to I. Therefore, this product takes us back from I to A and it equals A. E E E E is the inverse of A. However, This by itself tells us that p p 2 to convince us further that our original product is an inverse on the right as well, we multiply it on the right: A A E E2 E pe p E pe p E2E 2 p p p p 2 E E2 E p E p E2E I E E E E E E E E p If A is an invertible matrix and c 0 is a scalar, then: c ca A If A is an invertible matrix, then: A A T T Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 7

8 If A is an invertible matrix, then: A A n n So, now that we have learned how to construct these inverse matrices, what are we going to do with them? In the following chapters, we shall use inverse matrices in several contexts and situations, so they do have a wide range of applications. However, there is one for which you are ready and that links directly to the problem from which we got the idea of a matrix. Proof If Ax c is a linear system and A is a square matrix, then the system has a unique solution if, and only if A is invertible, On the other hand, if A is not invertible, its RREF has a zero row. By applying the Gauss-Jordan method to the system, we must end up with either a leading coefficient in the column of constants (no solution) or must have a free variable (infinitely many solutions). In either case, we do not have a unique solution. I can see from the proof that this also means that we can solve such a system by multiplying the vector of constants by the inverse function. Yes, and we could state that as an interesting fact. However, it will have little real uses for us, since in order to obtain the inverse we have to apply Gauss-Jordan, the same method we would use to solve the system, so we don t get any real advantage. But it is a little fact to keep in mind. And, to finish, here is another connection with things we have seen before. An n n matrix is invertible if and only if its rank is n. The proof of this one is a great exercise for you, so try it in the Learning questions. Time to digest the information of this section before we move on to deeper applications of them. If A is invertible, then, by multiplying by sides of the equation by its inverse we get: Ax c A Ax A c x A c But this means that only the vector is only one solution. A c satisfies the equation, so that there Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 8

9 Summary A matrix is invertible if, and only if its RREF is an identity matrix. To construct the inverse of an invertible matrix, we augment that matrix with the identity of the proper size and compute the RREF of such augmented matrix. The right half of this RREF is the inverse. Common errors to avoid Implement this method to compute inverses a sufficient number of times to understand how it works, but also reflect on all the properties used to obtain it and all the properties of inverses that stem from it. Learning questions for Section LA 4-7 Review questions:. Describe how to compute the inverse of an invertible matrix. 3. Explain how to obtain the inverse of the product of invertible matrices. 2. Identify the technical problems involved in the search for the inverse of a matrix. Memory questions:. Is an elementary matrix always invertible? 2. If a square matrix is invertible, what can be said about its RREF? 3. If A and B are invertible matrices, what is the inverse of AB? 4. How is the inverse of A used to solve the system Ax c? 5. If B is invertible, how many solutions does a system of the form Bx u have? 6. To what matrix do we apply Gauss-Jordan elimination when looking for A? Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 9

10 Computation questions: For each of the matrices provided in questions -2, compute its inverse, or explain why it does not exist, and if the inverse does exist, identify the first two elementary matrices you would use in its construction For each of the matrices presented in questions 3-8 determine the values of x for which the matrix is not invertible x x x 0 x x x Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 0

11 6. x 2 x 0 x 2 x 0 x x 7. sin x 2 0 sin x 2 0 sin x cos x 8. x x 3 0 x x 8 x 9. Given the matrices A 0 4, 2 B, compute A B. Then use this fact to determine the inverse of the matrix HINT: Look at how A and C are related C. 20. Given the matrices A B A and 0 2 B , check that 2. Write the system x2y 3 3x y 7 of the matrix of coefficients to solve the system. as a single matrix equation and use the inverse Theory questions:. Which matrices can be written as the product of elementary matrices? 2. Which visible feature of the matrix invertible? 0 b 0 3. Which triangular matrices are invertible? a 0 0 c d 0 tells you that it is not 4. If a square matrix A is not symmetric, by which matrix can it be multiplied so that the product is a symmetric matrix? 5. For which square matrices C is it true that CC C C? 6. If A and B are invertible matrices of the same dimensions, does it always follow that A B is also invertible? 7. What can we say about the RREF of a non-invertible square matrix? Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page

12 Proof questions:. Prove that if B is an invertible matrix, then the product AB commutes if and only if the product AB commutes. 2. Prove that if A is an invertible matrix and c 0 is a scalar, then ca A c 3. Prove that if A is an invertible matrix, then A A T T 4. Prove that if A is an invertible matrix, then A A n n 6. Prove that a 2 2 matrix is invertible if and only if its two rows are not parallel vectors. 7. Prove that an n n matrix is invertible if and only if its rank is n. 8. Use the Gauss-Jordan method to construct the inverse of the matrix / a / c valid. / b / d and identify all restrictions on a, b, c, d that make the answer 9. Prove that any square matrix can be written as the product of a set of elementary matrices and an upper triangular matrix. 5. Prove that the inverse of a symmetric matrix is also symmetric. Templated questions:. Construct a 22, 33 or 44 matrix and construct its inverse or show that it does not exist. What questions do you have for your instructor? Linear Algebra Chapter 4: Matrix Algebra Section 7: Inverse matrices Page 2

Number of solutions of a system

Number of solutions of a system Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 7 Number of solutions of a system What you need to know already: How to solve a linear system by using Gauss- Jordan elimination.

More information

Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3. Diagonal matrices

Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3. Diagonal matrices Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3 Diagonal matrices What you need to know already: Basic definition, properties and operations of matrix. What you

More information

Using matrices to represent linear systems

Using matrices to represent linear systems Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 4 Using matrices to represent linear systems What you need to know already: What a linear system is. What elementary operations

More information

Dependence and independence

Dependence and independence Roberto s Notes on Linear Algebra Chapter 7: Subspaces Section 1 Dependence and independence What you need to now already: Basic facts and operations involving Euclidean vectors. Matrices determinants

More information

Chapter 4. Solving Systems of Equations. Chapter 4

Chapter 4. Solving Systems of Equations. Chapter 4 Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.

More information

Roberto s Notes on Linear Algebra Chapter 11: Vector spaces Section 1. Vector space axioms

Roberto s Notes on Linear Algebra Chapter 11: Vector spaces Section 1. Vector space axioms Roberto s Notes on Linear Algebra Chapter 11: Vector spaces Section 1 Vector space axioms What you need to know already: How Euclidean vectors work. What linear combinations are and why they are important.

More information

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors Roberto s Notes on Linear Algebra Chapter 0: Eigenvalues and diagonalization Section Eigenvalues and eigenvectors What you need to know already: Basic properties of linear transformations. Linear systems

More information

Basic matrix operations

Basic matrix operations Roberto s Notes on Linear Algebra Chapter 4: Matrix algebra Section 3 Basic matrix operations What you need to know already: What a matrix is. he basic special types of matrices What you can learn here:

More information

Cofactors and Laplace s expansion theorem

Cofactors and Laplace s expansion theorem Roberto s Notes on Linear Algebra Chapter 5: Determinants Section 3 Cofactors and Laplace s expansion theorem What you need to know already: What a determinant is. How to use Gauss-Jordan elimination to

More information

Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2

Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2 Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch

More information

Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 4. Matrix products

Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 4. Matrix products Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 4 Matrix products What you need to know already: The dot product of vectors Basic matrix operations. Special types of matrices What you

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

1300 Linear Algebra and Vector Geometry

1300 Linear Algebra and Vector Geometry 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.

More information

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems

More information

Lecture 2 Systems of Linear Equations and Matrices, Continued

Lecture 2 Systems of Linear Equations and Matrices, Continued Lecture 2 Systems of Linear Equations and Matrices, Continued Math 19620 Outline of Lecture Algorithm for putting a matrix in row reduced echelon form - i.e. Gauss-Jordan Elimination Number of Solutions

More information

Roberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2. Orthogonal matrices

Roberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2. Orthogonal matrices Roberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2 Orthogonal matrices What you need to know already: What orthogonal and orthonormal bases for subspaces are. What you can learn here:

More information

Linear Algebra for Beginners Open Doors to Great Careers. Richard Han

Linear Algebra for Beginners Open Doors to Great Careers. Richard Han Linear Algebra for Beginners Open Doors to Great Careers Richard Han Copyright 2018 Richard Han All rights reserved. CONTENTS PREFACE... 7 1 - INTRODUCTION... 8 2 SOLVING SYSTEMS OF LINEAR EQUATIONS...

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row

More information

Basic methods to solve equations

Basic methods to solve equations Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:

More information

Row and column spaces

Row and column spaces Roberto s Notes on Linear Algebra Chapter 7: Subspaces Section 4 Row and column spaces What you need to know already: What subspaces are. How to identify bases for a subspace. Basic facts about matrices.

More information

Integration by inverse substitution

Integration by inverse substitution Roberto s Notes on Integral Calculus Chapter : Integration methods Section 9 Integration by inverse substitution by using the sine function What you need to know already: How to integrate through basic

More information

Review Solutions for Exam 1

Review Solutions for Exam 1 Definitions Basic Theorems. Finish the definition: Review Solutions for Exam (a) A linear combination of vectors {v,..., v n } is: any vector of the form c v + c v + + c n v n (b) A set of vectors {v,...,

More information

Topic 14 Notes Jeremy Orloff

Topic 14 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 Row reduction and subspaces 4. Goals. Be able to put a matrix into row reduced echelon form (RREF) using elementary row operations.. Know the definitions of null and column

More information

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example

More information

36 What is Linear Algebra?

36 What is Linear Algebra? 36 What is Linear Algebra? The authors of this textbook think that solving linear systems of equations is a big motivation for studying linear algebra This is certainly a very respectable opinion as systems

More information

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33 Linear Algebra 1/33 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct

More information

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3 Math 0 Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination and substitution

More information

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3 Linear Algebra Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

Inverses and Elementary Matrices

Inverses and Elementary Matrices Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose a 11 x 1 +a 12 x 2 + +a 1n x n = b 1 a 21 x 1 +a 22 x 2 + +a 2n x n = b 2 a n1 x

More information

Vector Spaces. 9.1 Opening Remarks. Week Solvable or not solvable, that s the question. View at edx. Consider the picture

Vector Spaces. 9.1 Opening Remarks. Week Solvable or not solvable, that s the question. View at edx. Consider the picture Week9 Vector Spaces 9. Opening Remarks 9.. Solvable or not solvable, that s the question Consider the picture (,) (,) p(χ) = γ + γ χ + γ χ (, ) depicting three points in R and a quadratic polynomial (polynomial

More information

CHAPTER 8: MATRICES and DETERMINANTS

CHAPTER 8: MATRICES and DETERMINANTS (Section 8.1: Matrices and Determinants) 8.01 CHAPTER 8: MATRICES and DETERMINANTS The material in this chapter will be covered in your Linear Algebra class (Math 254 at Mesa). SECTION 8.1: MATRICES and

More information

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression

More information

BASIC NOTIONS. x + y = 1 3, 3x 5y + z = A + 3B,C + 2D, DC are not defined. A + C =

BASIC NOTIONS. x + y = 1 3, 3x 5y + z = A + 3B,C + 2D, DC are not defined. A + C = CHAPTER I BASIC NOTIONS (a) 8666 and 8833 (b) a =6,a =4 will work in the first case, but there are no possible such weightings to produce the second case, since Student and Student 3 have to end up with

More information

Integration by partial fractions

Integration by partial fractions Roberto s Notes on Integral Calculus Chapter : Integration methods Section 15 Integration by partial fractions with non-repeated quadratic factors What you need to know already: How to use the integration

More information

[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]

[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.] Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the

More information

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34

CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34 Linear Algebra /34 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct axes)

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

2 Systems of Linear Equations

2 Systems of Linear Equations 2 Systems of Linear Equations A system of equations of the form or is called a system of linear equations. x + 2y = 7 2x y = 4 5p 6q + r = 4 2p + 3q 5r = 7 6p q + 4r = 2 Definition. An equation involving

More information

Elementary Linear Algebra

Elementary Linear Algebra Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We

More information

Matrices and RRE Form

Matrices and RRE Form Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

CHAPTER 7: TECHNIQUES OF INTEGRATION

CHAPTER 7: TECHNIQUES OF INTEGRATION CHAPTER 7: TECHNIQUES OF INTEGRATION DAVID GLICKENSTEIN. Introduction This semester we will be looking deep into the recesses of calculus. Some of the main topics will be: Integration: we will learn how

More information

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.

More information

February 20 Math 3260 sec. 56 Spring 2018

February 20 Math 3260 sec. 56 Spring 2018 February 20 Math 3260 sec. 56 Spring 2018 Section 2.2: Inverse of a Matrix Consider the scalar equation ax = b. Provided a 0, we can solve this explicity x = a 1 b where a 1 is the unique number such that

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Mon Feb Matrix algebra and matrix inverses. Announcements: Warm-up Exercise:

Mon Feb Matrix algebra and matrix inverses. Announcements: Warm-up Exercise: Math 2270-004 Week 5 notes We will not necessarily finish the material from a given day's notes on that day We may also add or subtract some material as the week progresses, but these notes represent an

More information

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =

More information

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of

( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they

More information

Linear Algebra, Summer 2011, pt. 2

Linear Algebra, Summer 2011, pt. 2 Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................

More information

Introduction to Matrices

Introduction to Matrices POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder

More information

Section 2.2: The Inverse of a Matrix

Section 2.2: The Inverse of a Matrix Section 22: The Inverse of a Matrix Recall that a linear equation ax b, where a and b are scalars and a 0, has the unique solution x a 1 b, where a 1 is the reciprocal of a From this result, it is natural

More information

Matrices MA1S1. Tristan McLoughlin. November 9, Anton & Rorres: Ch

Matrices MA1S1. Tristan McLoughlin. November 9, Anton & Rorres: Ch Matrices MA1S1 Tristan McLoughlin November 9, 2014 Anton & Rorres: Ch 1.3-1.8 Basic matrix notation We have studied matrices as a tool for solving systems of linear equations but now we want to study them

More information

Chapter 2. Matrix Arithmetic. Chapter 2

Chapter 2. Matrix Arithmetic. Chapter 2 Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the

More information

Inverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3

Inverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3 3., 3.3 Inverting Matrices P. Danziger 1 Properties of Transpose Transpose has higher precedence than multiplication and addition, so AB T A ( B T and A + B T A + ( B T As opposed to the bracketed expressions

More information

INVERSE OF A MATRIX [2.2]

INVERSE OF A MATRIX [2.2] INVERSE OF A MATRIX [2.2] The inverse of a matrix: Introduction We have a mapping from R n to R n represented by a matrix A. Can we invert this mapping? i.e. can we find a matrix (call it B for now) such

More information

For all For every For each For any There exists at least one There exists There is Some

For all For every For each For any There exists at least one There exists There is Some Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following

More information

Linear Algebra. Chapter Linear Equations

Linear Algebra. Chapter Linear Equations Chapter 3 Linear Algebra Dixit algorizmi. Or, So said al-khwarizmi, being the opening words of a 12 th century Latin translation of a work on arithmetic by al-khwarizmi (ca. 78 84). 3.1 Linear Equations

More information

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to : MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..

More information

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ] Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.

More information

Section Gaussian Elimination

Section Gaussian Elimination Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..

More information

7.6 The Inverse of a Square Matrix

7.6 The Inverse of a Square Matrix 7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to

More information

Differential Equations

Differential Equations This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is

More information

1300 Linear Algebra and Vector Geometry

1300 Linear Algebra and Vector Geometry 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Introduction: linear equations Read 1.1 (in the text that is!) Go to course, class webpages.

More information

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway

Linear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row

More information

March 19 - Solving Linear Systems

March 19 - Solving Linear Systems March 19 - Solving Linear Systems Welcome to linear algebra! Linear algebra is the study of vectors, vector spaces, and maps between vector spaces. It has applications across data analysis, computer graphics,

More information

Take the Anxiety Out of Word Problems

Take the Anxiety Out of Word Problems Take the Anxiety Out of Word Problems I find that students fear any problem that has words in it. This does not have to be the case. In this chapter, we will practice a strategy for approaching word problems

More information

The following are generally referred to as the laws or rules of exponents. x a x b = x a+b (5.1) 1 x b a (5.2) (x a ) b = x ab (5.

The following are generally referred to as the laws or rules of exponents. x a x b = x a+b (5.1) 1 x b a (5.2) (x a ) b = x ab (5. Chapter 5 Exponents 5. Exponent Concepts An exponent means repeated multiplication. For instance, 0 6 means 0 0 0 0 0 0, or,000,000. You ve probably noticed that there is a logical progression of operations.

More information

Lecture 9: Elementary Matrices

Lecture 9: Elementary Matrices Lecture 9: Elementary Matrices Review of Row Reduced Echelon Form Consider the matrix A and the vector b defined as follows: 1 2 1 A b 3 8 5 A common technique to solve linear equations of the form Ax

More information

MATH 310, REVIEW SHEET 2

MATH 310, REVIEW SHEET 2 MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,

More information

Integration by substitution

Integration by substitution Roberto s Notes on Integral Calculus Chapter : Integration methods Section 1 Integration by substitution or by change of variable What you need to know already: What an indefinite integral is. The chain

More information

Math 31 Lesson Plan. Day 5: Intro to Groups. Elizabeth Gillaspy. September 28, 2011

Math 31 Lesson Plan. Day 5: Intro to Groups. Elizabeth Gillaspy. September 28, 2011 Math 31 Lesson Plan Day 5: Intro to Groups Elizabeth Gillaspy September 28, 2011 Supplies needed: Sign in sheet Goals for students: Students will: Improve the clarity of their proof-writing. Gain confidence

More information

1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is

1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is Solutions to Homework Additional Problems. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. (a) x + y = 8 3x + 4y = 7 x + y = 3 The augmented matrix of this linear system

More information

MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)

MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and

More information

3.4 Elementary Matrices and Matrix Inverse

3.4 Elementary Matrices and Matrix Inverse Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary

More information

Polynomials; Add/Subtract

Polynomials; Add/Subtract Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x

More information

Algebra & Trig Review

Algebra & Trig Review Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The

More information

Definition of geometric vectors

Definition of geometric vectors Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 2 of geometric vectors What you need to know already: The general aims behind the concept of a vector. What you can learn here: The

More information

Systems of equation and matrices

Systems of equation and matrices Systems of equation and matrices Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 23, 2013 Warning This is a work in progress. I can not ensure it to be mistake free at the moment. It is also lacking

More information

SYDE 112, LECTURE 7: Integration by Parts

SYDE 112, LECTURE 7: Integration by Parts SYDE 112, LECTURE 7: Integration by Parts 1 Integration By Parts Consider trying to take the integral of xe x dx. We could try to find a substitution but would quickly grow frustrated there is no substitution

More information

Spanning, linear dependence, dimension

Spanning, linear dependence, dimension Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R have the same size (and so does 3-space, R 3 ) That is, there is a function between

More information

Methods for Solving Linear Systems Part 2

Methods for Solving Linear Systems Part 2 Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use

More information

A summary of factoring methods

A summary of factoring methods Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What

More information

Determinants and Scalar Multiplication

Determinants and Scalar Multiplication Properties of Determinants In the last section, we saw how determinants interact with the elementary row operations. There are other operations on matrices, though, such as scalar multiplication, matrix

More information

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.

base 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation. EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences

More information

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your

More information

Linear Algebra Handout

Linear Algebra Handout Linear Algebra Handout References Some material and suggested problems are taken from Fundamentals of Matrix Algebra by Gregory Hartman, which can be found here: http://www.vmi.edu/content.aspx?id=779979.

More information

Quadratic Equations Part I

Quadratic Equations Part I Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

More information

Elimination and back substitution

Elimination and back substitution Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 2 Elimination and back substitution What you need to know already: What a (linear) system is. What it means to solve such

More information

EXAM 2 REVIEW DAVID SEAL

EXAM 2 REVIEW DAVID SEAL EXAM 2 REVIEW DAVID SEAL 3. Linear Systems and Matrices 3.2. Matrices and Gaussian Elimination. At this point in the course, you all have had plenty of practice with Gaussian Elimination. Be able to row

More information

Math 250B Midterm I Information Fall 2018

Math 250B Midterm I Information Fall 2018 Math 250B Midterm I Information Fall 2018 WHEN: Wednesday, September 26, in class (no notes, books, calculators I will supply a table of integrals) EXTRA OFFICE HOURS: Sunday, September 23 from 8:00 PM

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

chapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS

chapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader

More information

MODEL ANSWERS TO THE THIRD HOMEWORK

MODEL ANSWERS TO THE THIRD HOMEWORK MODEL ANSWERS TO THE THIRD HOMEWORK 1 (i) We apply Gaussian elimination to A First note that the second row is a multiple of the first row So we need to swap the second and third rows 1 3 2 1 2 6 5 7 3

More information

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

CHAPTER 8: MATRICES and DETERMINANTS

CHAPTER 8: MATRICES and DETERMINANTS (Section 8.1: Matrices and Determinants) 8.01 CHAPTER 8: MATRICES and DETERMINANTS The material in this chapter will be covered in your Linear Algebra class (Math 254 at Mesa). SECTION 8.1: MATRICES and

More information