# 1 Determinants. Definition 1

Save this PDF as:

Size: px
Start display at page:

## Transcription

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 by the matrix. The determinant can also be used to find the solutions of linear systems and is therefore a helpful tool in matrix algebra. The determinant will be defined recursively, i.e. we will first define the determinant for a 2 2 matrix, then we will define the determinant of a n n matrix based on determinants of (n ) (n ) matrices. Applying these rules recursively will lead to the determinant. Definition If A is a 2 2 matrix [ ] a b A = c d then the determinant of A is defined by = A = ad cb. (Cofactor Expansion along the first row) If A is a square matrix of size n the = A = where the cofactor of the entry a ij is C ij defined as n a j C j j= C ij = ( ) i+j M ij where the minor of entry a ij is M ij, the determinant of the submatrix that remains after the ith row and jth column are deleted from A Example (a) Let then This was easy because A is a 2 2 matrix (b) Let A = [ = = ( 2) 5(5) = 27 A = ] Then the minor (and cofactor) of a is (delete row and column ) M = 2 3 = 2() 3( ) =, so C = ( ) + M =

2 The minor (and cofactor) of a 2 is (delete row and column 2) M 2 = 3 = () ( 3)( ) = 2, so C 2 = ( ) +2 M 2 = ( )( 2) = 2 The minor (and cofactor) of a 3 is (delete row and column 3) M 3 = = (3) ( 3)( 2) = 3, so C 3 = ( ) +3 M 3 = 3 With these = a C + a 2 C 2 + a 3 C 3 = () + 2(2) + 3( 3) = = 4 The definition is based on the cofactor expansion along the first row. One can prove that it is possible to expand along any row or column Theorem (a) Expansion along row i = n a ij C ij = a i C i + a i2 C i a in C in j= (b) Expansion along column j no proof. = n a ij C ij = a j C j + a 2j C 2j +... a nj C nj i= Example 2 The last theorem allows to make smart choice, when calculating a determinant. Let Let A = For finding the determinant expansion alon the 4th column looks really easy because = 5( ) all other entries are zero and do not provide any more terms. To find the minor M 4 it is easiest to expand along the third row = ( 5)(6)( ) = ( 5)6((3) ( 3)( 2)) = ( 5)6( 3) = 90 2

3 Definition 2 If A is a square matrix of size n and C ij is the cofactor of a ij, then the matrix C C 2... C n C 2 C C 2n... C n C n2... C nn is called the matrix of cofactors from A. The adjoint of A, adj(a), is defined to be the transpose of the matrix of cofactors for A. Example 3 Use A from example (b): then the cofactors of A are and Theorem 2 If A is an invertible matrix then A = C = C 2 = 2 C 3 = 3 C 2 = 7 C 22 = 0 C 23 = 9 C 3 = 4 C 32 = 4 C 33 = 4 adj(a) = A = adj(a) Example 4 Use matrix A from example 3, then 2 3 A =, adj(a) = and from ex. (b), we know = 4. The last theorem now let us know that A = Theorem 3 If A is an n n matrix (upper triangular, lower triangular, or diagonal), then is the product of the entries on the diagonal of the matrix, that is = a a 22 a nn 3

4 Proof: Expansion along row to row (column) n successively for lower (upper) triangular matrices shows the result. Example 5 (a) Let A = then according to the theorem = (0)( 4) = 40. That was easy. (b) Let Then det(b) = 0. B = In order to take advantage of this property we will see how we can use elementary row operations to transform a matrix into a upper triangular matrix to find the determinant.. Row Reductions to Find Determinants Theorem 4 Let A be a square matrix. If A has a row of zeros or a column of zeros, then = 0. Proof: Do cofactor expansion along the row(column) that is all zero and you find that the determinant has to be equal to zero. Theorem 5 Let A be a square matrix then = det(a T ). Proof: The determinant of A can be found by expansion along row this is equal to the cofactor expansion along column of the transposed matrix.. Theorem 6 Let A be a square matrix of size n. (a) If B is the matrix that results when a single row(column) of A is multiplied by k R, then det(b) = k. (b) If B is the matrix that results when two rows or two columns of A are interchanged, then det(b) =. (c) If B is the matrix that results when a multiple of one row(column) of A is added to another row(column), then det(b) = 4

5 From this theorem we see the effect of elementary row(column) operations on the determinant, this will help to find the determinant because now we can transform A into a upper or lower triangular form, and then easily find the determinant. Example 6 Find for A = Transform A into row echelon form = 3 3 = = 2 0 3/ Because of the theorem above we get Theorem 7 = / /2 = = 2()()(5/2) = 5 (a) If E results from multiplying a row of I n by k R, then det(e) = k. (b) If E results from interchanging two rows of I n, then det(e) =. (c) If E results from adding a multiple of one row of I n to another, then det(e) =. Example 7 (a) = 5, =, = Theorem 8 If A is a square matrix with two proportional columns(rows), then = 0. Proof: Using the elementary row operation of subtracting the multiple of one row (column) to another row (column) will transform A into a matrix with a row (column) of zeros, and this matrix for that reason has a determinant of zero. Since this elementary row operation does not change the determinant, the determinant of A must be 0. The next theorem shows how determinants can be used to find solutions of linear systems. 5

6 Theorem 9 Cramer s Rule If Ax = b is a linear system in n unknowns such that 0, then the system has a unique solution, which is x = det(a ), x 2 = det(a 2),..., x n = det(a n) where A j is the matrix obtained by replacing the entries in column j of A by the entries of the solution vector b b 2 b =. Proof: If 0 then A is invertible and x = A b is the unique solution of the linear system Using Theorem?? we get resulting in so If x = A b = b n adj(a)b = x = A j = C C 2... C n C 2 C C n2 b b n.. b 2.. C n C 2n... C nn b C + b 2 C b n C n b C 2 + b 2 C b n C n2. b C n + b 2 C 2n b n C nn x j = b C j + b 2 C 2j b n C nj a a 2... a j b a j+... a n a 2 a a 2j b 2 a j+... a 2n a n a n2... a nj b n a j+... a nn The cofactors of this matrix are equal to the cofactors of A for all entries in column j. I.e. when calculating the determinant of A j by expanding along column j, one gets substituting this into the equation above we find which concludes the proof. det(a j ) = b C j + b 2 C 2j b n C nj x j = det(a j) 6

7 Example 8 Cramer s Rule Assume the following is the augmented matrix of a linear system [ 5 ] 2 3 Using Cramer s Rule we can now give the solution right away: x = det(a ) 2() 3() = 5() () = 4, x 2 = det(a 2) = 5(3) (2) 5() () = Properties of The Determinant Theorem 0 Let k R, and A be a square matrix then Example 9 In general det(a + B) + det(b) Let [ 5 2 A = 3 then det(ka) = k n A + B = ] [ 2, B = 4 [ and = 3, det(b) = 9, det(a + B) = 8, so + det(b) det(a + B) There following theorem shows, when determinant can be added Theorem Let A, B, C square matrices of the same size, which only differ in a single row, say the rth row. Assume that the rth row of C is obtained by adding the corresponding entries in the rth row of A and B, then det(c) = + det(b) Example 0 Illustrate the previous theorem with this example. Let [ ] [ ] [ A =, B =, C = 3 4 ] ] then = 3, det(b) = 3, det(c) = 0, in this example truly as predicted by the theorem det(c) = + det(b). Theorem 2 A square matrix A is invertible if and only if 0 ] 7

8 Theorem 3 If A and B are square matrices of the same size then Theorem 4 If A is an invertible square matrix then det(ab) = det(b) det(a ) = Proof: Since A A = I, therefore det(a A) = det(i) =, therefore (because of Theorem??) det(a ) =, since A is invertible 0, and we find det(a ) =.3 A Combinatorial Approach to Determinants Observe that by expansion along the first row, we get a a 2 a 2 a 22 = a a 22 a 2 a 2 a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 = a a 22 a 33 a a 23 a 32 a 2 a 2 a 33 + a 2 a 3 a 23 + a 3 a 2 a 32 a 3 a 3 a 22 Find that in each term of the determinant there is exactly one entry from each row and each column. Also find that all possible terms are included with the calculation of the determinant. This can be used to find the determinant differently than by expansion along a row or column. Some of the terms are added some are subtracted, in order to determine the signs we will discuss permutations and their inversions. Definition 3 A permutation of the numbers {, 2,..., n} is an arrangement of these integers in some order without omission or repetition. Example (3,, 2) is a permutation of {, 2, 3}, but (, 3, 3) is not a permutation of {, 2, 3} because 2 is missing and 3 is repeated. There are n! different permutations of {, 2,..., n}. Definition 4 Let (j, j 2,..., j n } denote a permutation of {, 2,..., n}. An inversion is said to occur in (j, j 2,..., j n ), whenever a larger integer precedes a smaller one. 8

9 The number of inversion in the elementary products will determine the sign used in the determinant. The total number of inversions in (j, j 2,..., j n ) is best found by () Find the number of integers that are smaller than j that follow j (2) find the number of integers that are smaller than j 2 and follow j 2 Continue the counting process for the remaining entries j 3,..., j n. The sum of these numbers is the total number of inversions in the permutation. Example 2 (5, 2, 6, 3,, 4 has =9 inversions. (, 2, 3) has 0 inversions. Definition 5 A permutation is called odd(even), if the total number of inversions is an odd(even) integer. Definition 6 Let A be a square matrix, then an elementary product from A is any product of n entries from A, where no two come from the same row or column or a j a j2 a njn where (j, j 2,..., j n ) is a permutation of {, 2,..., n}. A signed elementary product is a j a j2 a njn if (j, j 2,..., j n ) is even a j a j2 a njn if (j, j 2,..., j n ) is odd Theorem 5 Let A be a square matrix, then is the sum of all signed elementary products Example 3. Let then A = [ = 5(3) (2) = 3 the number of inversions for the first term is 0 so, elementary product is even, and the number of inversions for the second term is the elementary product is odd. 2. Let Then the elementary products are A = ]

10 or develop along the last row, then products permutation inversions odd/even 5(3)(0) (,2,3) 0 even 5(4)(-2) (,3,2) odd 2()(0) (2,,3) odd 2(4)(-3) (2,3,) 2 even (-)()(-2) (3,,2) 2 even (-)(3)(-3) (3,2,) 3 odd = 0 ( 40) 0 + ( 24) = 9 = ( )( 2 + 9) 4( 0 + 6) = 9 0

### 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

### 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

### (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

### 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

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

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.

### 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)

### 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 +

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### ( % . 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

### 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

### 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

### 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

### 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)(

### 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:

### 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

### 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

### 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

### 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 =

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### 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,

### 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:

### = [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

### 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

### 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:

### 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:

### 1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

### 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

### 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

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### Chapter 1. Determinants

Chapter 1. Determinants This material is in Chapter 2 of Anton & Rorres (or most of it is there). See also section 3.5 of that book. 1.1 Introductory remarks The determinant of a square matrix A is a number

### LINEAR ALGEBRA. September 23, 2010

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

### 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,

### 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

### 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

### 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,

### 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

### Row Echelon Form and Reduced Row Echelon Form

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

### 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

### 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

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

### 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

### 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

### 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

### 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.

### 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

### Mathematics of Cryptography Part I

CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

### 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

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### 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

### 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

### 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

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Preliminaries of linear algebra

Preliminaries of linear algebra (for the Automatic Control course) Matteo Rubagotti March 3, 2011 This note sums up the preliminary definitions and concepts of linear algebra needed for the resolution

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

### 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

### 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

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved.

Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. 1 Sum, Product and Transpose of Matrices. If a ij with

### Matrix Calculations: Inverse and Basis Transformation

Matrix Calculations: Inverse and asis Transformation A. Kissinger (and H. Geuvers) Institute for Computing and Information ciences Intelligent ystems Version: spring 25 A. Kissinger (and H. Geuvers) Version:

### 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

### Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices

Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### 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

### 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

### Divisibility of the Determinant of a Class of Matrices with Integer Entries

Divisibility of the Determinant of a Class of Matrices with Integer Entries Sujan Pant and Dennis I. Merino August 8, 010 Abstract Let x i = n j=1 a ij10 n j, with i = 1,..., n, and with each a ij Z be

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### 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

### 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

### 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

### 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