ENGR-1100 Introduction to Engineering Analysis. Lecture 21. Lecture outline
|
|
- Todd Randall
- 5 years ago
- Views:
Transcription
1 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 of matrices. Determinants: Finding determinants by cofactor expansion along any row or column. S 1
2 Definition: row equivalent Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent. A Finite sequence of row operation B A Inverse of the row operation in reverse order B A is row equivalent to B Theorem 1 If A is an nxn matrix, then the following statements are equivalent, that is, if one is true then all are: (a) A is invertible (b) AX=0 has only the trivial solution (the only solution is x 1 =0, x 2 =0 x n =0) (c) A is row equivalent to I n. If A -1 exists then A Finite sequence of row operation I n S 2
3 (b) AX=0 has only the trivial solution (the only solution is x 1 =0, x 2 =0 x n =0) a 11 a 12. a 1n a 21 a 22. a 2n 0 Finite sequence : : : : : : : : of row operation : : : a n1 a n2. a nn x 1 =0, x 2 =0 x n =0 Theorem 2 If A is an nxn invertible matrix, then the sequence of row operations that reduces A to I n reduces I n to A -1. A The same sequence I n I n of row operation A -1 S 3
4 The procedure The same sequence A I n I n A -1 of row operation Example 1: Find the inverse of * 2 * * 2 S 4
5 * * * * Thus: A -1 = S 5
6 Statement: If the procedure used in this example is attempted on a matrix that is not invertible, then, it will be impossible to reduce the left side to I by row operation. At some point in the computation a row of zeros will occur on the left side, and it can be concluded that the given matrix is not invertible : : : In last example we found that A is invertible. What can we say about the following system of linear equations? (hint: use theorem 1) x 1 +2x 2 +3x 3 =0 2x 1 +5x 2 +3x 3 =0 x 1 +8x 3 =0 Answer: The system has only the trivial solution using the second statement in theorem 1 : x 1 =x 2 =x 3 =0 S 6
7 Class assignment 1: Find the inverse of the following matrices (if the matrices are invertible), and verify (for the invertible matrices) by multiplying the original matrix B= S 7
8 System of linear equations and invertibility S 8
9 Theorem 1 If A is invertible nxn matrix, then for each nx1 matrix B, the system of equation AX=B has exactly one solution, namely X= A -1 B. a 11 a 12. a 1n x 1 B 1 a 21 a 22. a 2n x 2 = B 2 : : : : : a n1 a n2. a nn x 1 x n a 11 a 12. a 1n -1 B n B 1 x 2 = a 21 a 22. a 2n B 2 : : : : : x n a n1 a n2. a nn B n Example: consider the system of linear equations: x 1 +2x 2 +3x 3 =5 2x 1 +5x 2 +3x 3 =3 x 1 +8x 3 =17 We can write the this system as AX=B x X= x 2 B= x 3 17 S 9
10 In example 1 we found that the inverse of A -1 is : A -1 = By theorem 1 the solution of the system is: X=A -1 B= = or: x 1 =1, x 2 =-1, x 3 =2 Why it is useful to find the inverse of a matrix Many problems in engineering and science involve systems of n linear equation with n unknown (that is square matrix). The method is particularly useful when it is necessary to solve a series of systems: AX=B 1, AX=B 2.. AX=B k, In this case each has the same square matrix A and the solutions are: X=A -1 B 1, X=A -1 B 2. X=A -1 B k S 10
11 Electronic circuitry AC=V current voltage a 11 a 12. a 1n c 1 v 1 a 21 a 22. a 2n c 2 = v 2 : : : : : a n1 a n2. a nn c n v n -1 c 1 a 11 a 12. a 1n v 1 c 2 = a 21 a 22. a 2n v 2 : : : : : c n a n1 a n2. a nn v n Theorem 3 If A is an nxn matrix, then the following statements are equivalent: (a) A is invertible (b) AX=0 has only the trivial solution (the only solution is x 1 =0, x 2 =0 x n =0) (c) A is row equivalent to I n. (d) AX=B is consistent for every nx1 matrix B. S 11
12 Class assignment 2: Solve the following system using the method you learned today in class x 1 +2x 2 +2x 3 =-1 x 1 +3x 2 + x 3 =4 x 1 +3x 2 + 2x 3 =3 S 12
13 S 13
14 Why study determinants? They have important applications to system of linear equations and can be used to produce formula for the inverse of an invertible matrix. The determinant of a square matrix A is denoted by det(a) or A. If A is 1x1 matrix [a 11 ] Then: det(a)= a 11 For example [-7] det (A)=det(-7)=-7 Determinant of a 2x2 matrix If A is a 2x2 matrix a 11 a 12 a 21 a 22 Then we define a 11 a 12 det(a)= a 21 a 22 = det(a)= a 11 a 22 -a 12 a 21 a 11 a 12 a 21 a 22 S 14
15 Example 1 Then we define det(a)= = det(a)= 5X2-4X3= Determinant of a 3x3 matrix If A is a 3x3 matrix a 11 a 12 a 13 a 21 a 22 a 23 Then we define a 11 a 12 a 13 det(a)= a 21 a 22 a 23 a 31 a 32 a 33 Example det(a)= = a 11 = 1 a 31 a 32 a 33 a 22 a 23 -a 12 a 21 a 23 +a 13 a 21 a 22 a 32 a 33 a 31 a 33 a 31 a =1[0*2 -(-1*2)] 5 [1*2-2*3] - 3[-1*1-0*3]=25 S 15
16 Minor and cofactor of a matrix entry Definition: If A is a square matrix, then the minor of entry a ij is denoted by M ij and is defined to be the determinant of submatrix that remains after the i th row and j th column are deleted from A. The number (-1) i+j M ij is denoted by C ij and is called the cofactor of entry a ij. For 3x3 matrix a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 M 11 = a 22 a 23 a 32 a 33 C 11 =(-1) 1+1 M 11 = M 11 Example The minor of a 11 is: The cofactor of a 11 is: 5 3 M 11 = 0 8 = 5X8-3X0=40 C 11 =(-1) i+j M ij =(-1) 2 M 11 =M 11 =40 The minor of a 23 is: M 23 = = 1X0-2X1=-2 The cofactor of a 23 is: C 23 =(-1) i+j M ij =(-1) 5 M 23 =-M 23 =2 S 16
17 Finding determinant using the cofactor a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 C = a 22 a C a a a 32 a 12 = - 33 a 31 a 33 C 13 = a 21 a 22 a 31 a 32 a 11 a 12 a 13 a 21 a 22 a 23 = a 11 C 11 +a 12 C 12 +a 13 C 13 a 31 a 32 a 33 And for nxn matrix: = a 11 C 11 +a 12 C a 1n C 1n Example: evaluate det(a) for: det(a) = a 11 C 11 +a 12 C 12 + a 13 C 13 +a 14 C det(a)=(1) - (-3) (0) = (1)(35)-0+(2)(62)-(-3)(13)=198 S 17
18 Theorem 1 The determinant of nxn matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i n and 1 j n, det(a)=a 1j C 1j +a 2j C 2j +..+a nj C nj Cofactor expansion along the j th column det(a)=a i1 C i1 +a i2 C i2 +..+a in C in Cofactor expansion along the i th row Example 5: evaluate det(a)= By a cofactor along the third column det(a)=a 13 C 13 +a 23 C 23 +a 33 C 33 det(a)= -3* (-1) *(-1) *(-1) = det(a)= -3(-1-0)+2(-1) 5 (-1-15)+2(0-5)=25 S 18
19 Class assignment 3: Let Find det(a) using (a) cofactor expansion along any row or column. S 19
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
More informationDeterminants 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
More informationMath 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
More informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More informationChapter 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
More informationDeterminants. 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
More informationMATRIX 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
More informationFormula 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
More informationLesson 3. Inverse of Matrices by Determinants and Gauss-Jordan Method
Module 1: Matrices and Linear Algebra Lesson 3 Inverse of Matrices by Determinants and Gauss-Jordan Method 3.1 Introduction In lecture 1 we have seen addition and multiplication of matrices. Here we shall
More informationElementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary
More information4. 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.
More informationLinear 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
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationChapter 2. Square matrices
Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a
More informationProperties of the Determinant Function
Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant:
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
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.
More informationComponents and change of basis
Math 20F Linear Algebra Lecture 16 1 Components and change of basis Slide 1 Review: Isomorphism Review: Components in a basis Unique representation in a basis Change of basis Review: Isomorphism Definition
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationMATRICES AND MATRIX OPERATIONS
SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)
More informationDeterminant of a Matrix
13 March 2018 Goals We will define determinant of SQUARE matrices, inductively, using the definition of Minors and cofactors. We will see that determinant of triangular matrices is the product of its diagonal
More informationdet(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
More informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More informationMath 320, spring 2011 before the first midterm
Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,
More informationDeterminants: 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
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationMATH 2030: EIGENVALUES AND EIGENVECTORS
MATH 2030: EIGENVALUES AND EIGENVECTORS Determinants Although we are introducing determinants in the context of matrices, the theory of determinants predates matrices by at least two hundred years Their
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationDeterminants 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 +...
More informationMATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.
MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationUndergraduate Mathematical Economics Lecture 1
Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses
More informationGraduate Mathematical Economics Lecture 1
Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists
More informationsum of squared error.
IT 131 MATHEMATCS FOR SCIENCE LECTURE NOTE 6 LEAST SQUARES REGRESSION ANALYSIS and DETERMINANT OF A MATRIX Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition You will now look
More information1 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
More informationLECTURE 12: SOLUTIONS TO SIMULTANEOUS LINEAR EQUATIONS. Prof. N. Harnew University of Oxford MT 2012
LECTURE 12: SOLUTIONS TO SIMULTANEOUS LINEAR EQUATIONS Prof. N. Harnew University of Oxford MT 2012 1 Outline: 12. SOLUTIONS TO SIMULTANEOUS LINEAR EQUATIONS 12.1 Methods used to solve for unique solution
More informationMAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.
MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices
More informationMethods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
More information1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)
1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix
More informationLecture 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-
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More information1 Last time: determinants
1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)
More informationMatrix 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
More informationLecture 8: Determinants I
8-1 MATH 1B03/1ZC3 Winter 2019 Lecture 8: Determinants I Instructor: Dr Rushworth January 29th Determinants via cofactor expansion (from Chapter 2.1 of Anton-Rorres) Matrices encode information. Often
More informationDeterminants and Scalar Multiplication
Properties of Determinants In the last section, we saw how determinants interact with the elementary row operations. There are other operations on matrices, though, such as scalar multiplication, matrix
More informationMATRIX DETERMINANTS. 1 Reminder Definition and components of a matrix
MATRIX DETERMINANTS Summary Uses... 1 1 Reminder Definition and components of a matrix... 1 2 The matrix determinant... 2 3 Calculation of the determinant for a matrix... 2 4 Exercise... 3 5 Definition
More informationMATRICES 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
More informationDeterminants of 2 2 Matrices
Determinants In section 4, we discussed inverses of matrices, and in particular asked an important question: How can we tell whether or not a particular square matrix A has an inverse? We will be able
More informationNotes 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
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationLectures on Linear Algebra for IT
Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 11. Determinants
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Michaelmas Term 2015 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Michaelmas Term 2015 1 / 10 Row expansion of the determinant Our next goal is
More informationHomework Set #8 Solutions
Exercises.2 (p. 19) Homework Set #8 Solutions Assignment: Do #6, 8, 12, 14, 2, 24, 26, 29, 0, 2, 4, 5, 6, 9, 40, 42 6. Reducing the matrix to echelon form: 1 5 2 1 R2 R2 R1 1 5 0 18 12 2 1 R R 2R1 1 5
More informationMATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
More informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationDeterminants. 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.
More informationMatrices. In this chapter: matrices, determinants. inverse matrix
Matrices In this chapter: matrices, determinants inverse matrix 1 1.1 Matrices A matrix is a retangular array of numbers. Rows: horizontal lines. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3
ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)-matrix Consider the following row operations on A (1) Swap the positions any
More informationDeterminants. 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,
More informationSolution Set 7, Fall '12
Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det
More informationa 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12
24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2
More informationEXAM. Exam #1. Math 2360, Second Summer Session, April 24, 2001 ANSWERS
i i EXAM Exam #1 Math 2360, Second Summer Session, 2002 April 24, 2001 ANSWERS i 50 pts. Problem 1. In each part you are given the augmented matrix of a system of linear equations, with the coefficent
More informationSPRING OF 2008 D. DETERMINANTS
18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants
More informationAnnouncements 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
More informationLinear Algebra Primer
Linear Algebra Primer D.S. Stutts November 8, 995 Introduction This primer was written to provide a brief overview of the main concepts and methods in elementary linear algebra. It was not intended to
More informationLecture 12 Simultaneous Linear Equations Gaussian Elimination (1) Dr.Qi Ying
Lecture 12 Simultaneous Linear Equations Gaussian Elimination (1) Dr.Qi Ying Objectives Understanding forward elimination and back substitution in Gaussian elimination method Understanding the concept
More informationChapter 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
More informationA matrix A is invertible i det(a) 6= 0.
Chapter 4 Determinants 4.1 Definition Using Expansion by Minors Every square matrix A has a number associated to it and called its determinant, denotedbydet(a). One of the most important properties of
More informationDeterminants and Cramer's Rule
eterminants and ramer's Rule This section will deal with how to find the determinant of a square matrix. Every square matrix can be associated with a real number known as its determinant. The determinant
More informationIntroduction 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.
More informationTOPIC 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:
More information7.3. Determinants. Introduction. Prerequisites. Learning Outcomes
Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering
More information9 Appendix. Determinants and Cramer s formula
LINEAR ALGEBRA: THEORY Version: August 12, 2000 133 9 Appendix Determinants and Cramer s formula Here we the definition of the determinant in the general case and summarize some features Then we show how
More informationChapter 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
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More information2 b 3 b 4. c c 2 c 3 c 4
OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a
More informationThe 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
More informationMath 60. Rumbos Spring Solutions to Assignment #17
Math 60. Rumbos Spring 2009 1 Solutions to Assignment #17 a b 1. Prove that if ad bc 0 then the matrix A = is invertible and c d compute A 1. a b Solution: Let A = and assume that ad bc 0. c d First consider
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationMATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:
More informationMATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS
MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on
More informationMath 18.6, Spring 213 Problem Set #6 April 5, 213 Problem 1 ( 5.2, 4). Identify all the nonzero terms in the big formula for the determinants of the following matrices: 1 1 1 2 A = 1 1 1 1 1 1, B = 3 4
More informationMatrix Calculations: Determinants and Basis Transformation
Matrix Calculations: Determinants and asis Transformation A. Kissinger Institute for Computing and Information ciences Version: autumn 27 A. Kissinger Version: autumn 27 Matrix Calculations / 32 Outline
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationSection 5.3 Systems of Linear Equations: Determinants
Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations
More informationMTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~
MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is One-to-one Onto Question
More informationInverses and Determinants
Engineering Mathematics 1 Fall 017 Inverses and Determinants I begin finding the inverse of a matrix; namely 1 4 The inverse, if it exists, will be of the form where AA 1 I; which works out to ( 1 4 A
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationPresentation by: H. Sarper. Chapter 2 - Learning Objectives
Chapter Basic Linear lgebra to accompany Introduction to Mathematical Programming Operations Research, Volume, th edition, by Wayne L. Winston and Munirpallam Venkataramanan Presentation by: H. Sarper
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationECON 186 Class Notes: Linear Algebra
ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).
More informationRow Space and Column Space of a Matrix
Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.
More information1 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
More information