# c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

Save this PDF as:

Size: px
Start display at page:

Download "c c c c c c c c c c a 3x3 matrix C= has a determinant determined by"

## Transcription

1 Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant. For example the areas of volumes of certain geometric figures can be found by calculating the determinant of the matrix made up of the vectors that form these geometric figures. We will also use the determinant to introduce a special value called the eigenvalue of the square matrix. The eigenvectors associated with the eigenvalue have the special property that when multiplied by the matrix they produce a parallel vector. Introduction to Determinants Any square nxn matrix A has a determinant. The notation is A or det(a). a 1x1 matrix A= [ a 11 ] has a determinant determined by its only term det(a) = a 11 a b a 2x2 matrix B= c d has a determinant determined by det(b) = ad bc c11 c12 c13 a 3x3 matrix C= c21 c22 c 23 has a determinant determined by c31 c32 c 33 det(c) = c 11 c 22 c 33 + c 12 c 23 c 31 + c 13 c 21 c 32 c 13 c 22 c 31 c 11 c 23 c 32 c 12 c 21 c 33 there is a pattern that makes this easy to memorize, its often called the basket weaving technique. c c c c c c c c c c c c c c c Ex: Calculate the determinants of the following matrices 7 5 A = 8 6 B= and C=

2 Applications of Determinants: Areas and Volumes 2 v= v, v w= w, w in R that are not parallel and share a starting vertex. Let [ ] and [ ] Then the area of the parallelogram formed by v and w if given by the determinant of v1 v2 w1 w 2 (take the absolute value since area should be non-negative) w v Let v= [ v, v, v ] and w= [ w, w, w ] and z= [ z, z, z ] in 3 R that are not coplanar and they all share the same initial point. Then the volume of the parallelepiped form by v, w, and z is given by the determinant of (again absolute value because volume) v1 v2 v3 w1 w2 w 3 z1 z2 z 3 z w v What happens when the vectors are parallel or coplanar? Calculating Determinants by Coplanar Expansion There is another method of finding determinants using elements of a row multiplied by determinants of square matrices created by certain elements in the other rows. Let A be a nxn matrix with n 2. The (i,j) th sub-matrix A ij of the matrix A is the (n)x(n) matrix obtained by deleting the i th row of A and the j th column of A. The (i,j) th minor, denoted M ij, is the determinant of the submatrix A ij of A. Let A be a nxn matrix with n 2. The (i,j) th cofactor of A, denoted C ij, is () i+j multiplied by the (i,j) th minor of A. In other words C ij = () i+j M ij So given an 3x3 matrix there are nine submatrices, nine minors, and nine cofactors. Ex: Find all nine submatrices, minors, and cofactors of A=

3 The Formal Definition of the Determinant Let A be an nxn matrix. The determinant of A, denoted det(a), is defined as follows: If n = 1, then det(a) = a 11 If n >1, then det(a) = a n1 C n1 + a n2 C n2 + a n3 C n3 + + a nn C nn When we calculate the determinant using the sum of cofactors, we call the procedure the cofactor expansion Ex: Given A= calculate the determinant by cofactor expansion 3 ways We could also define the determinant of a nxn matrix A with n >1 as det(a) = a 1n C 1n + a 2n C 2n + a 3n C 3n + + a nn C nn Choose the row/column with the most zeros to simplify the computation since any of them with produce the determinant. Ex: Calculate the determinant by cofactor expansion for the following A. A= B. B= Determinants by Row Reduction Determinant of Upper Triangular Matrices: Let A be an upper triangular nxn matrix then, det(a) = a 11 a 22 a 33.a nn (product of diagonal entries) Ex: calculate det(a) if A= What effect does row reduction have on the determinant? Let A be a nxn matrix with the det(a)=p and let c be a scalar, then: If the row operation is R i R j then the determinant becomes P If the row operation if R i = cr i the the determinant becomes cp If the row operation if R i = cr j +r i then the determinant stays the same Ex: Use row reduction to calculate the determinant of A= Theorem: A nxn matrix A is nonsingular iff det(a) 0 Corollary: Let A be an nxn matrix, then rank(a)=0 iff det(a) 0 Assume A is a nxn matrix, then the following are equivalent

5 Cramer s Rule : Let AX = B represent a system of n linear equations in n variables such that det(a) 0 then the solu on to the system is det( A1 ) det( A2 ) det( An ) x1 = x2 = xn = where the i th column of A i is the column of constants in the system of equations, (B). 5x 3y 10z= 9 Ex: solve by Cramer Rule 2x+ 2y 3z= 4 3x y + 5z = 1 Eigenvalues and Diagonalization We will define eigenvalues and eigenvectors for matrices in order to find when possible the diagonal form of a square matrix. Let A be a nxn matrix. A real number λ is an eigenvalue of A iff there exists any nonzero vector X for which AX = λx. Also, any nonzero vector X for which AX = λx is called an eigenvector corresponding to the eigenvalue λ. Let A be a nxn matrix and λ be an eigenvalue of A then, the set Eλ { xax λx} called the eigenspace corresponding to the eigenvalue λ. = = is The Characteristic Polynomial of a Matrix A We need procedure to determine all eigenvalues and eigenvectors of an nxn matrix A. If X is an eigenvector for A corresponding to the eigenvalue λ then AX = λx = λi n X or (λi n A)X = 0 Therefore X is a nontrivial solution to the homogeneous system whose coefficient matrix is (λi n A). Let A be a nxn matrix and λ be a real number then λ is an eigenvalue of A iff det(λi n A) = 0 The eigenvectors corresponding to λ are the nontrivial solutions X to the homogeneous system (λi n A)X = 0. The eigenspace E λ corresponding to λ us the set of all nontrivial solutions to (λi n A)X = 0 Let A be a nxn matrix then the characteristic polynomial of A is the polynomial given by P A (λ) = det (λi n A) The real roots of the characteristic polynomial are the eigenvalues of A.

6 Procedure to find Eigenvalues, Eigenvectors, and Eigenspace: 1. State the characteristic polynomial by solving the determinant P A (λ) = det (λi n A) 2. The values λ 1, λ 2, λ k that are the roots to P A (λ) are the eigenvalues 3. Set up the homogeneous system (λi n A)X = 0 where the coefficient matrices are (λi n A) for each λ 1, λ 2, λ k. The nontrivial solution(s) to this system is the eigenvector corresponding to each eigenvalue. 4. All linear combinations of the eigenvectors for each λ 1, λ 2, λ k make up the eigenspace corresponding to each eigenvalue and eigenvector. State E λk. Ex: Find all the eigenvalues of the given matrix then state the eigenvectors associated with each eigenvalue and the corresponding eigenspace a. A= b. B= Diagonalization We now seek to take a square matrix and turn it into a representative diagonal matrix by use of eigenvalues and eigenvectors. Realize if we can replace a matrix with a diagonal matrix then we will greatly simplify any computations involving the original matrix. Therefore our next goal is to present a formal method for using eigenvalues and eigenvectors to find a diagonal form of a square matrix, provided it exists. A matrix A is similar to a matrix A if there exists some nonsingular matrix P such that A = P AP. Properties of Similar Matrices Let A, B and C be nxn matrices. then the following properties are true. 1. A is similar to A 2. If A is similar to B, then B is similar to A. 3. If A is similar to B and B is similar to C then A is similar to C. 4. If A and B are similar they have the same eigenvalues. An nxn matrix A is diagonalizable when A is similar to a diagonal matrix. That is, A is diagonalizable when there exists an invertible matrix P such that D = P AP where D is a diagonal matrix. An nxn matrix A is diagonalizable iff it has n linearly independent eigenvectors. Let A and P be nxn matrices such that each column of P is an eigenvector of A. If P is nonsingular then D = P AP is a diagonal matrix similar to the matrix A. The i th main diagonal entry d ii of D is the eigenvalue for the eigenvector forming the i th column of P.

7 Notice in the two examples we did for finding eigenvalues and eigenvectors, we see if we form P in either case, then P doesn t exist and hence, the original matrices are not diagonalizable. Method for Diagonalizing a nxn matrix A (if possible) 1. Calculate the characteristic polynomial and find all the real roots, λ 1, λ 2, λ k. In other words find all λ such that if P(λ) = det (λi n A) then P(λ) = For each eigenvalue λ m in λ 1, λ 2, λ k we find the corresponding eigenvectors. If there are n λ s that are distinct, then A is diagonalizable and if there are less than n λ s, then A is not diagonalizable. 3. If there are n λ s that are distinct then there are n eigenvectors. Form a matrix P such that the columns of P are eigenvectors and find P. 4. Check your work by checking that P AP is a diagonal matrix D whose main diagonal entries are the corresponding eigenvalues for the eigenvectors that formed that column Ex: Find the diagonal matrix D such that D = P AP for A=

### Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates

### 4. Determinants.

4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

### and let s calculate the image of some vectors under the transformation T.

Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

### Determinants by Cofactor Expansion (III)

Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...

### Math 240 Calculus III

The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

### Determinants Chapter 3 of Lay

Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j

### Lecture 10: Determinants and Cramer s Rule

Lecture 0: Determinants and Cramer s Rule The determinant and its applications. Definition The determinant of a square matrix A, denoted by det(a) or A, is a real number, which is defined as follows. -by-

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### TOPIC III LINEAR ALGEBRA

[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal

### 1 Determinants. 1.1 Determinant

1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

### Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

### 1. General Vector Spaces

1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### ANSWERS. E k E 2 E 1 A = B

MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,

### Matrix Operations: Determinant

Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant

### Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n

### A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

### MA 265 FINAL EXAM Fall 2012

MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators

### Determinants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262

Determinants Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations

### Introduction to Determinants

Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.

### Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Linear Algebra Primer

Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................

### Linear Algebra- Final Exam Review

Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.

### ENGR-1100 Introduction to Engineering Analysis. Lecture 21

ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility

### ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex

### Formula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column

Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the

### Chapter 4. Determinants

4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining

### Introduction to Matrices

214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the

### Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) Eigenvalues and Eigenvectors Fall 2015 1 / 14 Introduction We define eigenvalues and eigenvectors. We discuss how to

### Math 110 Linear Algebra Midterm 2 Review October 28, 2017

Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections

### Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

### MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

### ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

### Chapter 2: Matrices and Linear Systems

Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers

### Determinants. Beifang Chen

Determinants Beifang Chen 1 Motivation Determinant is a function that each square real matrix A is assigned a real number, denoted det A, satisfying certain properties If A is a 3 3 matrix, writing A [u,

### Lecture 2: Eigenvalues and their Uses

Spectral Graph Theory Instructor: Padraic Bartlett Lecture 2: Eigenvalues and their Uses Week 3 Mathcamp 2011 As you probably noticed on yesterday s HW, we, um, don t really have any good tools for finding

### Definition (T -invariant subspace) Example. Example

Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

### Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

### det(ka) = k n det A.

Properties of determinants Theorem. If A is n n, then for any k, det(ka) = k n det A. Multiplying one row of A by k multiplies the determinant by k. But ka has every row multiplied by k, so the determinant

### Linear Algebra and Vector Analysis MATH 1120

Faculty of Engineering Mechanical Engineering Department Linear Algebra and Vector Analysis MATH 1120 : Instructor Dr. O. Philips Agboola Determinants and Cramer s Rule Determinants If a matrix is square

### Eigenvalues and Eigenvectors. Review: Invertibility. Eigenvalues and Eigenvectors. The Finite Dimensional Case. January 18, 2018

January 18, 2018 Contents 1 2 3 4 Review 1 We looked at general determinant functions proved that they are all multiples of a special one, called det f (A) = f (I n ) det A. Review 1 We looked at general

### Eigenvalues, Eigenvectors, and Diagonalization

Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will

### MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.

MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ

### Extra Problems for Math 2050 Linear Algebra I

Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as

### Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains

Linear Algebra: Linear Systems and Matrices - Quadratic Forms and Deniteness - Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems

### EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko

### Introduction to Matrix Algebra

Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

### 1. Select the unique answer (choice) for each problem. Write only the answer.

MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

### Math 313 (Linear Algebra) Exam 2 - Practice Exam

Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2 - Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if

### Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant

### THE ADJOINT OF A MATRIX The transpose of this matrix is called the adjoint of A That is, C C n1 C 22.. adj A. C n C nn.

8 Chapter Determinants.4 Applications of Determinants Find the adjoint of a matrix use it to find the inverse of the matrix. Use Cramer s Rule to solve a sstem of n linear equations in n variables. Use

### Linear Algebra in Actuarial Science: Slides to the lecture

Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations

### MTH 102A - Linear Algebra II Semester

MTH 0A - Linear Algebra - 05-6-II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )

### 1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )

Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical

### Math 2331 Linear Algebra

5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu,

### 4. Linear transformations as a vector space 17

4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation

### MATRICES The numbers or letters in any given matrix are called its entries or elements

MATRICES A matrix is defined as a rectangular array of numbers. Examples are: 1 2 4 a b 1 4 5 A : B : C 0 1 3 c b 1 6 2 2 5 8 The numbers or letters in any given matrix are called its entries or elements

### MATH 369 Linear Algebra

Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

### 1 9/5 Matrices, vectors, and their applications

1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric

### Math 215 HW #9 Solutions

Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

### Announcements Wednesday, October 25

Announcements Wednesday, October 25 The midterm will be returned in recitation on Friday. The grade breakdown is posted on Piazza. You can pick it up from me in office hours before then. Keep tabs on your

### Math 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:

Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your

### MATH 1210 Assignment 4 Solutions 16R-T1

MATH 1210 Assignment 4 Solutions 16R-T1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,

### Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

### Topic 15 Notes Jeremy Orloff

Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.

### (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

### Notes on Determinants and Matrix Inverse

Notes on Determinants and Matrix Inverse University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 1 Definition of determinant Determinant is a scalar that measures the magnitude or size of

### Lecture Notes: Eigenvalues and Eigenvectors. 1 Definitions. 2 Finding All Eigenvalues

Lecture Notes: Eigenvalues and Eigenvectors Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Definitions Let A be an n n matrix. If there

### Examples and MatLab. Vector and Matrix Material. Matrix Addition R = A + B. Matrix Equality A = B. Matrix Multiplication R = A * B.

Vector and Matrix Material Examples and MatLab Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square

### Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

### Determinants: summary of main results

Determinants: summary of main results A determinant of an n n matrix is a real number associated with this matrix. Its definition is complex for the general case We start with n = 2 and list important

### Math 225 Linear Algebra II Lecture Notes. John C. Bowman University of Alberta Edmonton, Canada

Math 225 Linear Algebra II Lecture Notes John C Bowman University of Alberta Edmonton, Canada March 23, 2017 c 2010 John C Bowman ALL RIGHTS RESERVED Reproduction of these lecture notes in any form, in

### EE263: Introduction to Linear Dynamical Systems Review Session 5

EE263: Introduction to Linear Dynamical Systems Review Session 5 Outline eigenvalues and eigenvectors diagonalization matrix exponential EE263 RS5 1 Eigenvalues and eigenvectors we say that λ C is an eigenvalue

### EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only non-zero entries are on the diagonal This is equivalent to

### Math 4 - Summary Notes

Math 4 - Summary Notes July, 0 Chat: what is class about? Mathematics of Economics, NOT Economics! Lecturer NOT Economist - have some training, but reluctant to discuss strict economic models because much

### Determinants. 2.1 Determinants by Cofactor Expansion. Recall from Theorem that the 2 2 matrix

CHAPTER 2 Determinants CHAPTER CONTENTS 21 Determinants by Cofactor Expansion 105 22 Evaluating Determinants by Row Reduction 113 23 Properties of Determinants; Cramer s Rule 118 INTRODUCTION In this chapter

### 1 Matrices and Systems of Linear Equations

March 3, 203 6-6. Systems of Linear Equations Matrices and Systems of Linear Equations An m n matrix is an array A = a ij of the form a a n a 2 a 2n... a m a mn where each a ij is a real or complex number.

### Eigenvalues and Eigenvectors

CHAPTER Eigenvalues and Eigenvectors CHAPTER CONTENTS. Eigenvalues and Eigenvectors 9. Diagonalization. Complex Vector Spaces.4 Differential Equations 6. Dynamical Systems and Markov Chains INTRODUCTION

### Determinants. Recall that the 2 2 matrix a b c d. is invertible if

Determinants Recall that the 2 2 matrix a b c d is invertible if and only if the quantity ad bc is nonzero. Since this quantity helps to determine the invertibility of the matrix, we call it the determinant.

### The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices

The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative

### A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

### Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.

Quiz 1) Simplify 9999 999 9999 998 9999 998 2) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown Solving Linear Algebraic Equa3ons Basic Concepts Here only real

### ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact.

ORIE 6334 Spectral Graph Theory September 8, 2016 Lecture 6 Lecturer: David P. Williamson Scribe: Faisal Alkaabneh 1 The Matrix-Tree Theorem In this lecture, we continue to see the usefulness of the graph

### Linear algebra and applications to graphs Part 1

Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces

### Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,

### CHAPTER 6. Direct Methods for Solving Linear Systems

CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to

### 0.1 Eigenvalues and Eigenvectors

0.. EIGENVALUES AND EIGENVECTORS MATH 22AL Computer LAB for Linear Algebra Eigenvalues and Eigenvectors Dr. Daddel Please save your MATLAB Session (diary)as LAB9.text and submit. 0. Eigenvalues and Eigenvectors

### Elementary Linear Algebra Review for Exam 3 Exam is Friday, December 11th from 1:15-3:15

Elementary Linear Algebra Review for Exam 3 Exam is Friday, December th from :5-3:5 The exam will cover sections: 6., 6.2, 7. 7.4, and the class notes on dynamical systems. You absolutely must be able

### SOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4.

SOLUTIONS: ASSIGNMENT 9 66 Use Gaussian elimination to find the determinant of the matrix det 1 1 4 4 1 1 1 1 8 8 = det = det 0 7 9 0 0 0 6 = 1 ( ) 3 6 = 36 = det = det 0 0 6 1 0 0 0 6 61 Consider a 4

### Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O

### ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization

ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 21 Symmetric matrices An n n

### 8-15. Stop by or call (630)

To review the basics Matrices, what they represent, and how to find sum, scalar product, product, inverse, and determinant of matrices, watch the following set of YouTube videos. They are followed by several

### Math 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants

THE ALGEBRA OF DETERMINANTS 1. Determinants We have already defined the determinant of a 2 2 matrix: det = ad bc. We ve also seen that it s handy for determining when a matrix is invertible, and when it

### 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,

### Solutions to Final Exam 2011 (Total: 100 pts)

Page of 5 Introduction to Linear Algebra November 7, Solutions to Final Exam (Total: pts). Let T : R 3 R 3 be a linear transformation defined by: (5 pts) T (x, x, x 3 ) = (x + 3x + x 3, x x x 3, x + 3x