Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Size: px
Start display at page:

Download "Determinants. Dr. Doreen De Leon Math 152, Fall 2015"

Transcription

1 Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A. Definition. An elementary matrix, typically denoted E, is a matrix that is obtained by performing a single elementary row operation on the identity matrix. As such, there are three types of elementary matrices.. The first type is formed by exchanging row i of I with row j of I, which the author denotes as E i,j. 2. The second type is formed by multiplying row i of I by a nonzero scalar α, which the author denotes E i (α). 3. The third type is formed by adding a nonzero multiple α of row i to row j, which the author denotes E i,j (α). Example: For each of the following, determine if it is an elementary matrix (a) 0 0 (b) 0 0 (c) (d) Solution: 0 0 (e) (a) Yes. Row is formed by adding row 3 of the identity matrix to row.

2 (b) No. Rows and 3 of I are exchanged, and then the resulting rows and 2 are exchanged. (c) Yes. Row of the identity matrix is multiplied by 3. (d) No. Row is multiplied by 3 and then twice row 3 of the identity matrix is added to row. (e) Yes. Row 2 is formed by adding -2 times row of the identity matrix to row 2. We have the following theorem. textbook. Note that this is a summary of Theorem EMDRO in the Theorem. If an elementary row operation is performed on an m n matrix A, the resulting matrix can be written as EA, where the m m elementary matrix E is created by performing the same row operation on I m. Elementary matrices have another useful property. Theorem 2. If E is an elementary matrix, then E is nonsingular. Proof. The idea is that we can row reduce E to the identity matrix by reversing the row operation that formed E. If E = E i,j, then exchanging rows i and j again will give the identity matrix. If E = E i (α), multiply row i of E by to obtain the identity matrix. Finally, α if E = E i,j (α), perform the row operation that multiplies row i by α and adds it to row j. Therefore, each elementary matrix is row equivalent to the identity matrix and is thus nonsingular. In fact, we have the following useful theorem. Theorem 3. Suppose that A is a nonsingular matrix. Then there exist elementary matrices E, E 2,..., E t so that A = E t E t E 2 E. Proof. Since A is nonsingular, it is row equivalent to the identity matrix. Therefore, there is a sequence of t row operations that converts I to A. For each of these row operations, form the associated elementary matrix and denote these matrices by E, E 2,..., E t. Applying the first row operation to I gives the matrix E I. The second row operation gives E 2 (E I) = E 2 E I. and so on. The result of the full sequence of t operations will yield A, so we have A = E t E t E 2 E I = E t E t E 2 E. 2

3 Definition of the Determinant We need a few definitions first. Definition. Suppose that A is an m n matrix. Then the submatrix A ij (denoted A(i j) in the textbook) is the (m ) (n ) matrix obtain from A by removing row i and column j. Examples: Given 3 2 A = 0 4 5, [ ] 0 5 A 2 = 2 0 [ ] 3 A 23 =. 2 4 Exercise: Given find A 2 and A 34. Solution: A = , A 2 = 2, A 34 = Definition. Suppose A is an n n matrix. Then its determinant, det(a) = A, is an element of C defined recursively by. If n =, then det(a) = a. 2. If n 2, then det(a) = a det(a ) a 2 det(a 2 ) + + ( ) +n a n det(a n ) n = ( ) +j a j det(a j ). i= 2 5 Example: Find det(a) for A =

4 Solution: Using the formula given in the definition, we have = ( )+ () ( )+2 (2) ( )+3 (5) = 3[( ) + ( 3) + ( ) +2 (7) 0 ] 2[( ) + (0) + ( ) +2 (7) ] + +5[( ) + (0) 0 + ( ) +2 ( 3) ] = 3( 3 0) 2( 7) + 5(3) = 26. Note that this definition also leads to the standard formula for the determinant of a 2 2 matrix. [ ] a b Theorem 4. Let A =. Then det(a) = ad bc. c d Definition. Given an n n matrix A, the (i, j) cofactor of A, denoted C ij, is given by C ij = ( ) i+j det(a ij ). Then, using this, we can define det(a) = a C + a 2 C a n C n. () Equation () is a cofactor expansion across the first row. Computing Determinants There are a number of ways to compute the determinant. Theorem 5. The determinant of an n n matrix A can be computed by a cofactor expansion across any row i, det(a) = ( ) i+ a i det(a i ) + ( ) i+2 a i2 det(a i2 ) + + ( ) i+n a in det(a in ), called a cofactor expansion along row i, or down any column j, det(a) = ( ) +j a j det(a j ) + ( ) i+2 a 2j det(a 2j ) + + ( ) n+j a nj det(a nj ), called a cofactor expansion along column j. Example: Compute det(a), where A = ,

5 using a cofactor expansion. Solution: We choose to do a cofactor expansion along row 2. det(a) = ( ) 2+ a 2 det(a 2 ) + ( ) 2+2 a 22 det(a 22 ) + ( ) 2+3 a 23 det(a 23 ) + ( ) 2+4 a 24 det(a 24 ) 2 2 = 0 det(a 2 ) + 0 det(a 22 ) + ( ) det(a 24) 2 2 = We will use a cofactor expansion along row 3 to compute this determinant. ( det(a) = 3 ( ) ( ) ) 2 6 = 3[5( 0 ( 2)) + 4( 6 ( 4))] = 6. Another property of the determinant follows. Theorem 6. Suppose that A is a square matrix. Then det(a t ) = det(a). Example: Let Then, 3 4 A = det(a) = 0 + ( ) det(a 22 ) + 0 = = 2((2) 4(0)) = 4. And, since 0 0 A t = 2 2, det(a t ) = ( ) = (2(2) (0)) = 4 = det(a). Consider the following example. 5

6 Example: Let Find det(a). Solution: A = = ( ) = ( ) = ()( ) = ()(6)( ) = ()(6)(2)(7(5) 2(0)) = ()(6)(2)(7)(5) = 420. Notice that this is equivalent to multiplying the numbers on the diagonal of A. We can generalize this to the theorem following this requisite definition. Definition.. An upper triangular matrix is a square matrix whose entries below the main diagonal are zero. 2. A lower triangular matrix is a square matrix whose entries above the main diagonal are zero. 3. A diagonal matrix is a square matrix whose entries above and below the main diagonal are zero. Theorem 7. If A is a triangular matrix, then det(a) is the product of the entries on the main diagonal. 6

7 Example: Evaluate Solution: = ()(6)(2)(7)(5) = Properties of Determinants of Matrices Theorem 8. Suppose that A is a square matrix with a row with every entry a zero or a column with every entry a zero. Then det(a) = 0. Proof. Suppose that A is an n n matrix and that every entry in row i is 0. Then, we can find det(a) by doing a cofactor expansion along row i, giving det(a) = = = n ( ) i+j a ij det(a ij ) j= n ( ) i+j 0 det(a ij ) j= n 0 = 0. j= The proof for the case of a column consisting entirely of zeros is similar. Theorem 9 (Row Operations and the Determinant). Let A be a square matrix. (a) If a multiple of one row of A is added to another row to produce a matrix B, then det B = det(a). (b) If two rows of A are interchanged to produce B, then det B = det(a). (c) If one row of A is multiplied by a nonzero scalar α to produce B, then det B = α det(a). 7

8 2 4 Example: Let A = We form matrix B by r 2 2r 0 4 = B Then, det(a) = 0 C C C 33 = 5( ) = 5( 4) = 20. det(b) = ( 4)(5) = 20. So, we see that det(b) = 20 = det(a). 2 2 Example: Let A = We form matrix B as follows: r 2r = B Then, det(a) = (3)(6) = 9 2 and det(b) = (3)(6) = 8. So, we see that det(b) = 2 det(a). 2 3 Example: Let A = 0. We form matrix B by r 2 r Then, So, we see that det(b) = det(a). det(a) = ()(5) = 5. det(b) = 0 C C C 23 = 5( ) = 5. Note that as a consequence of the above properties, we can show the following. 8

9 Theorem 0. Suppose that A is a square matrix with two equal rows or two equal columns. Then det(a) = 0. Proof. Suppose A is an n n matrix such that rows i and j are equal. Let B be the matrix from by subtracting row i from row j. Then det(a) = det(b) = 0. Example: Compute det(a) by row reducing to echelon form, where A = Solution: = r 2 r 2 +2r r 3 r 3 3r r 4 r 4 r = r 3 27 r 3 = 27()()()(4) = = r 3 r 3 +4r 2 r 4 r 4 +3r 2 = r 4 r 4 8r 3 Determinants, Row Operations, and Elementary Matrices First, we will prove a few theorems about elementary matrices, which we will use to prove an important theorem in a bit. Theorem. For every n, det(i n ) =. Proof. We can see that I n is a triangular matrix. Therefore, det(i n ) is equal to the product of the entries on the diagonal, or det(i n ) = (n times) =. 9

10 Theorem 2 (Determinants of Elementary Matrices). For the three possible versions of an elementary matrix, we have the determinants () det(e i,j ) =, (2) det(e i (α)) = α, (3) det(e i,j (α)) =. This theorem is proved by using the fact that each elementary matrix is obtained by performing a single elementary row operation on the identity matrix, and then applying the theorem on elementary row operations and the determinant Theorem 3. If E is an elementary matrix, then det(ea) = det(e) det(a). The proof of this theorem uses the theorem on determinants of elementary matrices and the fact that if we let B = EA, then B is the matrix formed by performing the row operation from E on A. A Quick Note on Column Operations (not in text) We can perform column operations on a matrix in the same way we perform row operations. Column operations have the same effect on determinants as row operations. NOTE: Do NOT perform column operations when solving systems of equations. Determinants, Nonsingular Matrices, Matrix Multiplication Theorem 4. Let A be a square matrix. Then A is singular if and only if det(a) = 0. This is proved in the text, so I will prove the following equivalent theorem. Theorem 5. A square matrix A is nonsingular if and only if det(a) 0. Proof. A can be reduced to reduced row echelon form U with a finite number of row operations, so U = E k E k E A, where E i represents an elementary matrix. 0

11 Then, det(u) = det(e k E k E A) = det(e k ) det(e k ) det(e ) det(a). Since det(e i ) 0 for all i, det(a) = 0 if and only if det(u) = 0. If A is nonsingular, then U = I, so det(u) = = det(a) 0. If det(a) = 0, then det(u) = 0 = U contains a row consisting entirely of zeros (since det U = u u 22 u nn ). Therefore, A is singular. Theorem 6 (Nonsingular Matrix Equivalences, Round 5). Suppose that A is a square matrix. Then the following are equivalent. () A is nonsingular. (2) A row reduces to the identity matrix. (3) The null space of A contains only the zero vector (i.e., N (A) = {0}). (4) The linear system LS(A, b) has a unique solution for every b. (5) The columns of A are linearly independent. (6) A is invertible. (7) The column space of A is C n (Col(A) = C n ). (8) The determinant of A is nonzero, i.e., det(a) 0. Finally, we have the following property of determinants. Theorem 7. If A and B are n n matrices, then det(ab) = det(a) det(b). Proof. If A or B is singular, then so is AB. So, det(ab) = 0 = det(a) det(b). If A is nonsingular, then Therefore, we have, A = E E 2 E t. det(ab) = det(e E 2 E t B) = det(e ) det(e 2 ) det(e t ) det(b) = det(e E 2 E t ) det(b) = det(a) det(b).

12 Corollary. Let A be an n n matrix. Then det(a k ) = [det(a)] k for k a nonnegative integer Example: Let A = Find det(a 3 ) Solution: Since det(a 3 ) = [det(a)] 3, det(a) = 2(9)(2) = 36. det(a 3 ) = [36] 3 = Example: Show that if A is invertible, then Solution: Since A is invertible, det(a ) = det(a). AA = I = det(aa ) = det(i) = (det(a))(det(a) ) = = det(a ) = det(a). 3 Cramer s Rule We first need the following notation. For an n n matrix A and any b R n, let A i (b) be the matrix obtained from A by replacing column i of A by the vector b; so, A i (b) = [ a a i b a i+ a n ]. Theorem 8 (Cramer s Rule). Let A be an invertible n n matrix. For any b R n, the unique solution x of Ax = b has entries given by x i = det(a i(b)), for i =, 2,..., n. det(a) Proof. Let A = [ a a 2 a n ] and I = [ e e 2 e n ]. If Ax = b, then AI i (x) = A [ e e i x e i+ e n ] = [ Ae Ae i Ax Ae i+ Ae n ] = [ Ae Ae i b Ae i+ Ae n ] = [ a a i b a i+ a n ] = Ai (b). 2

13 Then, (det(a))(det(i i (x)) = det(a i (b), and x i det(a) = det(a i (b)) = x i = det(a i(b)). det(a) Note that det(a) 0 since A is invertible. Example: Use Cramer s rule to solve 2x + x 2 = 7 3x + x 3 = 8 x 2 + 2x 3 = 3. Solution: For this problem, 2 0 A = 3 0 and det(a) = Applying Cramer s rule, we have x = 4 = 2 4 = x 2 = 4 = 4 4 = x 3 = 4 = 4 4 =. 3 So, x =. 3

14 Use Cramer s Rule for Engineering Applications Systems of first order differential equations solved using Laplace transforms can lead to systems of equations like 6sx + 4x 2 = 5 9x + 2sx 2 = 2. We need to know (a) for what values of s the is solution unique, and (b) what the solution is for these values of s. First, we know from Cramer s rule that if the coefficient matrix A is invertible, the solution is unique. Since [ ] 6s 4 A =, 9 2s we have 6s 4 9 2s = 2s2 36 = 2(s 2 3) = 2(s + 3)(s 3). Therefore, the system has a unique solution if s ± 3. For such an s, we have s x = A = 0s + 8 2(s 2 3). 6s x 2 = A 2s 45 = 2(s 2 3). A Formula for A Cramer s rule leads to a general formula for the inverse of an n n matrix as follows. The j th column of A is a vector x that satisfies Ax = e j. The i th entry of x is the (i, j) th entry of A. 4

15 By Cramer s rule, the We can show that (i, j) th entry of A = x i = det(a i(e j )). det(a) det(a i (e j )) = ( ) i+j det(a ji ) = C ji, where C is the cofactor matrix (the matrix of cofactors of A), and So, A ji = (n ) (n ) matrix formed by deleting row j and column i of A. A = det(a) C C 2 C n C 2 C 22 C n2... = det(a) Ct. C n C 2n C nn The matrix C t is the adjugate of A, adj(a). This leads us to the following theorem. Theorem 9. Let A be an invertible n n matrix. Then A = det(a) adj(a) Example: Find A if A = Solution: First, note that det(a) = 3()(2) = 6. Then, C = ( ) = 2 C 2 = ( ) = 2 C 3 = ( ) = 3 ( 2) = C 2 = ( ) = 0 C 22 = ( ) = 6 C 23 = ( ) = 9 C 3 = ( ) = 0 C 32 ( ) = 0 C 33 = ( ) = 3 5

16 So, A = =

1 Determinants. Definition 1

1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors 2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information

2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant 2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i. Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

More information

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

Unit 18 Determinants

Unit 18 Determinants Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of

More information

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

More information

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

More information

( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

( % . This matrix consists of $ 4 5  5' the coefficients of the variables as they appear in the original system. The augmented 3  2 2 # 2  3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

More information

Matrix Inverse and Determinants

Matrix Inverse and Determinants DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

1.5 Elementary Matrices and a Method for Finding the Inverse

1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

More information

5.3 Determinants and Cramer s Rule

5.3 Determinants and Cramer s Rule 290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

Lecture 4: Partitioned Matrices and Determinants

Lecture 4: Partitioned Matrices and Determinants Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying

More information

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

More information

MATH36001 Background Material 2015

MATH36001 Background Material 2015 MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

More information

MATH10212 Linear Algebra B Homework 7

MATH10212 Linear Algebra B Homework 7 MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 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

More information

Elementary Row Operations and Matrix Multiplication

Elementary Row Operations and Matrix Multiplication Contents 1 Elementary Row Operations and Matrix Multiplication 1.1 Theorem (Row Operations using Matrix Multiplication) 2 Inverses of Elementary Row Operation Matrices 2.1 Theorem (Inverses of Elementary

More information

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, Chapter 1: Linear Equations and Matrices MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

More information

Using row reduction to calculate the inverse and the determinant of a square matrix

Using row reduction to calculate the inverse and the determinant of a square matrix Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible

More information

Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8. Matrices II: inverses. 8.1 What is an inverse? Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

Problems. Universidad San Pablo - CEU. Mathematical Fundaments of Biomedical Engineering 1. Author: First Year Biomedical Engineering

Problems. Universidad San Pablo - CEU. Mathematical Fundaments of Biomedical Engineering 1. Author: First Year Biomedical Engineering Universidad San Pablo - CEU Mathematical Fundaments of Biomedical Engineering 1 Problems Author: First Year Biomedical Engineering Supervisor: Carlos Oscar S. Sorzano September 15, 013 1 Chapter 3 Lay,

More information

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

More information

Matrices: 2.3 The Inverse of Matrices

Matrices: 2.3 The Inverse of Matrices September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.

More information

Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations. Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

More information

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that 0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

More information

Inverses and powers: Rules of Matrix Arithmetic

Inverses and powers: Rules of Matrix Arithmetic Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

More information

The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

More information

Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014 Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of

More information

Chapter 5. Matrices. 5.1 Inverses, Part 1

Chapter 5. Matrices. 5.1 Inverses, Part 1 Chapter 5 Matrices The classification result at the end of the previous chapter says that any finite-dimensional vector space looks like a space of column vectors. In the next couple of chapters we re

More information

9 Matrices, determinants, inverse matrix, Cramer s Rule

9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

More information

Chapter 4: Binary Operations and Relations

Chapter 4: Binary Operations and Relations c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

More information

Mathematics Notes for Class 12 chapter 3. Matrices

Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

More information

DETERMINANTS TERRY A. LORING

DETERMINANTS TERRY A. LORING DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation By-Product The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas

More information

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

More information

Determinants. Chapter Properties of the Determinant

Determinants. Chapter Properties of the Determinant Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

More information

1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.

1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. Math Exam - Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that

More information

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A. APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Lecture 10: Invertible matrices. Finding the inverse of a matrix

Lecture 10: Invertible matrices. Finding the inverse of a matrix Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices

More information

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015

Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015 Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number

More information

Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse. 1 Inverse Definition Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

EC9A0: Pre-sessional Advanced Mathematics Course

EC9A0: Pre-sessional Advanced Mathematics Course University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

More information

Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number

Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc. 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

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

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, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created:

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

In this leaflet we explain what is meant by an inverse matrix and how it is calculated.

In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 5.5 Introduction The inverse of a matrix In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 1. The inverse of a matrix The inverse of a square n n matrix A, is another

More information

Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

More information

Elementary Matrices and The LU Factorization

Elementary Matrices and The LU Factorization lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

Matrices, transposes, and inverses

Matrices, transposes, and inverses Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A = Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

More information

MAT Solving Linear Systems Using Matrices and Row Operations

MAT Solving Linear Systems Using Matrices and Row Operations MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented

More information

Solving Linear Systems, Continued and The Inverse of a Matrix

Solving Linear Systems, Continued and The Inverse of a Matrix , Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold: Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication). MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix

More information

Row Operations and Inverse Matrices on the TI-83

Row Operations and Inverse Matrices on the TI-83 Row Operations and Inverse Matrices on the TI-83 I. Elementary Row Operations 2 8 A. Let A =. 2 7 B. To interchange rows and 2 of matrix A: MATRIX MATH C:rowSwap( MATRIX NAMES :[A],, 2 ) ENTER. 2 7 The

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

We seek a factorization of a square matrix A into the product of two matrices which yields an

We seek a factorization of a square matrix A into the product of two matrices which yields an LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

Matrices, Determinants and Linear Systems

Matrices, Determinants and Linear Systems September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

Introduction to Matrix Algebra I

Introduction to Matrix Algebra I Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

More information

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67 Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of

More information

1 Eigenvalues and Eigenvectors

1 Eigenvalues and Eigenvectors Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

More information

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

More information