# Topic 1: Matrices and Systems of Linear Equations.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 of solving systems of linear equations, as well as the geometric interpretation of the solution set 1 Matrices and Determinants Rank of a matrix Matrix multiplication and inverse of a matrix An n m matrix is a set of n m real numbers arranged in n rows and m columns A matrix A of order n m is represented as follows a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm that is the element a ij is the entry in the i th row and the j th column We write A M n m Sometimes, we denote the matrix that has elements a ij using an older convention: A = (a ij i=1,,n j=1,,m or A = (a ij ij The elements a ii are called the diagonal of A When a matrix has n rows and m columns we say that the matrix is of the dimension (size n m If the number of rows and columns is the same (ie, if m = n, the matrix is called a square matrix The square matrix is known as the identity matrix of dimension n, and is denoted by I n Denition Given a matrix a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm M n m we dene the transpose matrix A t M m n (or A of A, as the matrix whose row i is equal to column i of A That is, a 11 a 21 a n1 A t = A a 12 a 22 a n2 = M m n a 1m a 2m a nm 1

2 2 Denition Given a square matrix, a 11 a 12 a 1n a 21 a 22 a 2n A = a n1 a n2 a nn the trace of A is the following real number M n m trace(a = a 11 + a a nn 11 Addition and multiplication by scalars Matrices of the same dimension can be added The sum is obtained by adding the elements that are located in the same row and column of the two matrices Thus, if A = (a ij i=1,,n j=1,,m and B = (b ij i=1,,n j=1,,m (as you see the matrices have the same dimension, then A + B = (a ij + b ij i=1,,n j=1,,m Example ( ( A = B = ( ( A + B = = ( ( Multiplication of a matrix by a real number is dened similarly A = (a ij i=1,,n j=1,,m and λ R, then λa = (λa ij i=1,,n j=1,,m Example Let λ be any (real number and consider the matrix ( A = then If we x λ = 7 7A = λa = ( ( λ 2 λ 1 λ 3 λ 9 λ 6 λ 5 = ( The matrix addition and multiplication by scalars satisfy the following properties that can be easily derived from the above denitions: Proposition 1 Let A, B and C be two matrices of the same dimension and let α and β be any real numbers Then, (1 A + B = B + A (commutativity (2 A + (B + C = (A + B + C (associativity (3 α(a + B = αa + αb (4 (α + βa = αa + βa (5 α(βa = (αβa (6 (A + B t = A t + B t If

3 3 12 Matrix Multiplication Consider now A = (a ij i=1,n j=1,,m an n m matrix and B = (b ij of size m l That is, the number of columns of matrix A equals the number of rows in matrix B This is a necessary (and sucient condition for being able to calculate the product matrix A B Denition The product matrix C = A B is the matrix with n rows (same number as A and l columns (same as B such that the element c ij = a i1 b 1j + a i2 b 2j + + a im b mj that is, the element of the product matrix in the position i, j is obtained by multiplying row i of A times column j of B Example Consider the matrices A = ( B = the matrix C = A B is the 2 2 matrix ( 49 8= ( C = Remark In general, the product of matrices is not commutative In fact, in many cases it is only possible to calculate one product, but not the other For instance, if we consider the matrices ( ( A = B = we can calculate A B But, B A is not dened, since the number of columns of B does not coincide with the number of rows of A If we take then, A = A B = ( ( B = B A = ( ( Proposition 2 If the product AB exists, then the product B t A t is also possible and (AB t = B t A t 13 Equivalent Matrices The Gauss-Jordan elimination Method Given a matrix A we say that B is equivalent to A if we can transform A into B using any combination of the following elementary operations: Multiply a row of A by any real number not equal to zero Interchange two rows Add a row of A to any other row These three operations can be described by means of matrix products Multiplying the i th row of an n m matrix by a real number a is equivalent to multiplying on the left by the identity matrix I n in which the element in the position ii is replaced by a

4 4 For instance, if we consider a matrix and want to multiply the 2nd row by 5 we have to multiply this matrix by To swap two rows, say rows i and j, the only thing we have to do is to multiply on the left by an identity matrix with rows i and j swapped For instance, if in the previous matrix: we want to exchange rows 1 and 3, then we multiply The operation of adding the j th row multiplied by a real number a, to the i th row is equivalent to multiply on the left by the identity matrix but with an a in the entry ij For instance, if in the matrix we want to add to row 2 seven times row 3, we can do it in the following way, that is, we put a 7 into the position 2, Denition 3 We say that a matrix A M n m is in (row echelon form if it satises the following: (1 All zero rows are at the bottom of the matrix (2 For each row i = 1,, n, if the rst non-zero element is a ij (that is, a ij 0 and a ik = 0 for every k < j, then a l,k = 0 for every l > i and every k l It follows that, if the matrix A is in echelon form, then given any row i, if its rst element that is not equal to zero is a ij then a kl = 0 whenever k > i and l j In an echelon matrix, every row (except perhaps the rst one starts with 0 and each row i + 1 starts with at least one more 0 than the preceding row i

5 5 For example, are matrices in echelon form While, is not The Gauss-Jordan elimination method gives us a systematic way of obtaining an echelon form for any matrix by means of elementary matrix operations We will explain the method by means of an example Consider the following matrix, (1 Reorder the rows so that all the rows with all elements equal to zero, if any, are below all other rows (2 Look for the rst column that doesn't have all zeros (3 Reorder the rows again, so that all the zeros in this column are below all the non-zero elements (4 If the matrix is already in echelon form, we are nished

6 6 (5 If not, we look for the rst element that violates the echelon condition (we call it pivotal and from now on forget all the rows above it (6 Repeat the previous step (ignoring the upper rows that have been already dealt with If the matrix is already in echelon form, we are done (7 If not, eliminate all the non-zero numbers that are in the same column as the pivotal element (let it be a ij and below it by adding multiples of row i to the multiples of rows below it: 3 row 3 5 row 2 = (8 If the matrix is already in echelon form, we are done Otherwise, repeat the operations with the rows that are still not echelon form, until you are done 14 Determinants To any square matrix one can assign a real number called the determinant of the matrix As we are going to see, this number is important and useful We shall dene it by induction Let A be a 1 1 matrix, ie it has a single row and column In other words, A is a scalar A = (a The determinant in this case is simply dened to be a Let A be a 2 2 matrix Then, the determinant is dened as det(a = a b c d = ad cb Let A be a 3 3 matrix In this case we have two equivalent ways of dening the determinant:

7 7 (1 Sarrus rule: a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11a 22 a 33 +a 12 a 23 a 31 +a 13 a 21 a 32 a 31 a 22 a 13 a 32 a 23 a 11 a 33 a 21 a 12 (2 Expanding by a row (or column: a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 23 a 32 a 33 a 21 a 12 a 13 a 32 a 33 + a 31 a 12 a 13 a 22 a 23 We don't have to be using the rst row or column; any row or column would work We have to be careful, though, with signs, since the term multiplied also by the element a ij has to be multiplied by ( 1 i+j Example If we want to expand the determinant by column 2, = ( ( ( = 2 ( (0 (1 1 = 5 For matrices of higher dimensions the rule is the same as for the 3 3 matrices: we can expand by rows or columns in such a way that the determinant of a 4 4 matrix can be calculated by calculating 4 determinants of 3 3 matrices In general, if A = (a ij is a square n n matrix, the minor complementary to the element a ij of A is the determinant of the n 1 n 1 submatrix that is obtained by eliminating the row i and the column j from the matrix A The cofactor A ij of the element a ij of A is the minor complementary to a ij multiplied by the factor ( 1 i+j With this notation, we can write the expansion of the determinant of a matrix A by row i as, or by column j, A = a i1 A i1 + a i2 A i2 + + a im A im A = a 1j A 1j + a 2j A 2j + + a nj A nj Example Consider the determinant It is best to expand this determinant by row 4 or column 3 This is so, because there are more zeroes in them So that in the end, we would have to do fewer calculations Let us expand by column 3: = ( ( As you see from this example, the more zeroes there are, the easier it is to do the calculations We shall now explain how to make there as many zeroes as possible

8 8 Proposition 4 (1 Let A and B be square matrices of the same size Then, det(a B = det(a det(b (2 If all elements of a row or a column of a matrix are equal to zero, then the determinant also equals to zero The remaining properties are easily derived from those above and can be used to increase the number of zeros Proposition 5 (1 If we multiply a row or a column of a matrix by a number, then the determinant is also multiplied by the same number (2 If we swap two rows (or two columns the determinant changes sign (3 If we add to a row (or a column a multiple of another row (respectively, column the determinant does not change (4 If two rows or columns are equal, then the determinant is equal to 0 Using the rules above, we can simplify the computation of determinants by using the same elementary operations we have used to put the matrix into its echelon form Example Consider the determinant: = = = = = = = = Rank of a Matrix Remark The echelon form obtained by the above procedure is not unique That is, when we apply the method above to reduce a matrix to an echelon form, the nal matrix that we obtain, depends on the exact way the steps are carried out However, one can prove that the number of zero rows obtained is independent of the procedure to nd the echelon form Denition 6 Given any matrix A we dene rank of A to be the number of rows not all equal to zero that the matrix has in any of its echelon forms Example Consider the matrix A =

9 9 it is equivalent to the echelon matrix: = so that the rank of A is four There is an equivalent denition of the matrix rank that uses determinants Proposition 7 Let A be any matrix The rank of A is the dimension of the largest square matrix with a non-zero determinant that we can construct from the original matrix by eliminating rows and columns Example (1 Take a matrix ( The rank of this matrix can't be more than 2 since it is impossible to construct inside it a 3 3 matrix Let us see if we can construct a 2x2 matrix with a non-zero determinant = 0 doesn't work = 1 0 Since, this determinant does not equal to zero, the rank of this matrix is 2 (2 If we now take ( then, as before, the rank can't be more than 2 Nor can it be smaller than 1, since for that the matrix would have to be a matrix of zeroes Let us see if it is 2: = 0, = 0, = 0 Therefore, the rank is 1 Remark The rank of two equivalent matrices is the same 16 Inverse Matrix Given an n n square matrix A, we say that it has an inverse if there exists an n n matrix, which we denote by A 1, such that A 1 A = AA 1 = I n, where I n is the identity matrix of order n n Remark Is there a unique inverse matrix? In other words, is it possible that there are two dierent matrices, say B and C such that BA = AB = I n and CA = AC = I n? Let us show that, if these equations hold then, we must have B = C Since, CA = I n, multiplying by B on the right we get that (CAB = I n B = B

10 10 and since, we see that B = C (CAB = C(AB = CI n = C Proposition 8 A square matrix has an inverse if and only if its determinant is not equal to zero Equivalently, an n n square matrix has an inverse if and only if it has the rank n (that is, it has full rank Therefore, using the properties of determinants it is easy to derive the following: Proposition 9 If A has an inverse, then det(a 1 = 1 det(a Proof: Since, A A 1 = I n, we must have that det(a A 1 = det(i n One checks easily that det(i n = 1 And, since det(a A 1 = det(a det(a 1 we must have that det(a det(a 1 = 1 and the proposition follows Proposition 10 Let A and B be n n square matrices Then, A B and B A have inverses only if A and B have inverses Furthermore, (A B 1 = B 1 A 1 (B A 1 = A 1 B 1 There are various methods of computing the inverse of a matrix, but undoubtedly the easiest is the Gauss-Jordan elimination method that we have used to put matrices in echelon form Consider the matrix a 11 a 12 a 1n a 21 a 22 a 2n A = a n1 a n2 a nn Construct the new matrix a 11 a 12 a 1n a 21 a 22 a 2n a n1 a n2 a nn To this entire matrix we apply the elementary operations until the left side becomes an identity matrix It can be shown that this can be done if and only if the original matrix has full-rank, in which case we obtain b 11 b 12 b 1n b 21 b 22 b 2n b n1 b n2 b nn and the matrix on the right is the inverse of A This method works since the matrix that we obtain on the right is a product of all those matrices by which we were multiplying the original matrix to turn it into the identity matrix

11 11 Example Consider the matrix construct the larger matrix: Make the following operations (f 3 f (f 3 +f At this point we see that this is a rank 3 matrix, so it can be inverted (2f 2 f (2f 1 f Finally, dividing by 2 Therefore, the inverse matrix is /2 1/2 1/ /2 1/2 1/ /2 1/2 1/2 A 1 = 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 For those who are absolutely in love with formulas, there does exist a formula that provides and inverse of a matrix For a square matrix A, its adjoint matrix (Adj(A is the matrix of cofactors of A That is, an element in the i th row and j th column of Adj(A, is A ij Proposition If A 0, then A 1 = 1 A (Adj(At 2 Systems of Linear Equations A system of linear equations is a system of equations of the form a 11 x a 1n x n = b 1 a m1 x a mn x n = where a ij and b k are known real numbers and x 1,, x n are the unknowns (or variables of the system b m

12 12 A system of linear equations can be rewritten using matrix notation as: a 11 a 1n x 1 b 1 = a m1 a mn x n b m A solution of the system is an n dimensional vector that solves the matrix equation Or, in other words, it is the vector of real numbers (x 1, x n which satisfy all the equations of the system Denition 11 A system of linear equations is called consistent if it has a solution, otherwise it is called inconsistent or overdetermined If the system of linear equations has multiple (actually, innitely many solutions, it is called underdetermined Example The system of linear equations { 2x + y = 5 4x + 2y = 7 doesn't have a solution, so it is inconsistent The system of equations { x + y = 5 4y = 8 has as its unique solution the vector (3, 2 Thus, it is consistent and uniquely determined The system of equations { x + y = 4 2x + 2y = 8 has innitely many solutions, since all vectors of the form (x, 4 x solve it; therefore, it is consistent and underdetermined 21 The Rouché-Frobenius Theorem The Rouché-Frobenius theorem gives us a criterion to decide when the system is consistent or inconsistent And, in the former case, how many parameters do we need to describe the set of solutions For instance, consider the system of equations { x + y = 2 2x + 2y = 4 It can be easily seen that the second equation is just the double of the rst, and is, therefore, redundant Thus, the set of solutions is the set of two-dimensional vectors (x, y that satisfy the condition y = 2 x If we choose a value of x we, therefore, obtain the unique value of y that satises the system This set, {(x, 2 x : x R} can be described by means of a single parameter We say that it has one degree of freedom The Rouché-Frobenius theorem tells us exactly how many parameters (degrees of freedom we need to describe a solution of any system Consider the system of linear equations,

13 13 a 11 x a 1n x n = b 1 a m1 x a mn x n = with m equations and n unknowns We dene the matrix of the system to be a 11 a 1n A = a m1 a mn The extended matrix of the system is a 11 a 1n b 1 (A b = a m1 a mn b m Theorem 1 (Rouché-Frobenius (1 The system is consistent if and only if rank A = rank(a b (2 Suppose the system is consistent (Hence, rank A = rank(a b n Then, (a The system has a unique solution if and only if rank A = rank(a b = n (b The system is underdetermined if and only if rank A = rank(a b < n In this case, the number of parameters necessary to describe the solutions of the system is n rank(a b m Example Consider the system x + y + 2z = 1 2x + y + 3z = 2 3x + 2y + 5z = 3 Its extended matrix is Calculate the rank of A and of (A b: Therefore, the rank of A and of (A b is 2 Hence, the system is consistent and underdetermined Proposition 12 A homogeneous system (ie, a system in which all free terms on the right hand side of the equation are equal to zero is always consistent (It has at least one solution

14 14 22 Gauss-Jordan elimination method for solving systems of equations The Rouché-Frobenius theorem tells us when the system of linear equations has (or does not have a solution Combining it with the Gauss-Jordan elimination method we can explicitly obtain the solutions The idea is that, if the two systems have equivalent extended matrices, they have the same solutions On the other hand, if a matrix is already put into its echelon form, it is very easy to calculate the solutions Consider, for example, the system whose extended matrix is The associated system has a unique solution To nd the solution we only have to recall that each column corresponds to a variable and that the column after the line corresponds to free terms To give the solution, let us calculate the variables from the bottom up The last row tells us that 2z = 4, so we can substitute z = 2 into the second equation and get y + 8 = 2, so that y = 6 Finally, we substitute these values into the rst equation to obtain that x = 1 Hence, x = 9 Let us now consider the system represented by the matrix: This system is consistent and underdetermined, since the rank of A and of the augmented matrix are both equal to 2 and the number of unknowns is 3 Therefore, we have one degree of freedom The second equation is y +4z = 2, which we rewrite as y = 2 4z Substituting now into the rst equation, we obtain that x + 3(2 4z + 5z = 1, meaning that x = 5 + 7z Therefore, the set of all solutions of our system is {(7z 5, 2 4z, z : z R} Sometimes we have to be careful with the variables we leave as parameters In the previous case any one of the variables would work ne In the following example that's not the case Here the second equations tells us that y = 1 So, y cannot be left as a parameter If, on the other hand we substitute now the value y = 1 into the rst equation, we obtain that x + 4 z = 1, so that x = z 3 Here, the variables that can be left as parameters are x or z The general solution of the system can be written as or, if you so prefer, {(z 3, 1, z R 3 : z R} {(x, 1, x + 3 R 3 : x R}

15 15 The reason that systems represented with equivalent matrices have the same solutions are the following If we multiply an equation by a real number dierent from zero, the solution doesn't change If we reorder the equations, the solution of the system doesn't change If we add (a multiple of one of the equations to another the solution doesn't change Thus, if two systems are represented by (row equivalent matrices, their solutions are the same The Gauss-Jordan elimination method consists of transforming a system into another one, representable by a matrix in echelon form The solutions of the new system are easy to calculate and are the same as those of the original system 23 Cramer's Rule Suppose now we have a system of n equations in n unknowns In this case the system can be represented by a square matrix This system has a unique solution if and only if it has full-rank, which, in turn, is equivalent to having a non-zero determinant of the matrix Cramer's rule provides a way of nding solutions of the system using the determinants This is how it works Consider the system: a 11 x a 1n x n = b 1 and suppose that a n1 x a nn x n = b n a 11 a 12 a 1n 0 a n1 a n2 a nn so the system has a unique solution Denote the unique solution of the system as (x 1, x 2,, x n Then, b 1 a 12 a 1n a 11 b 1 a 1n x b n a n2 a nn 1 = x a 11 a 12 a 1n a n1 b n a nn 2 = a 11 a 12 a 1n a n1 a n2 a nn a n1 a n2 a nn a 11 a 12 b 1 x a n1 a n2 b n n = a 11 a 12 a 1n a n1 a n2 a nn We shall make two observations about his method

16 16 Cramer's method requires substantially more operations than the Gauss- Jordan elimination method It is easy to extend Cramer's method to underdetermined systems (we won't do this in this class, though

### (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

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### 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 =

### 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:

### 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,

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### 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

### 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

### ( % . 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

### 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

### 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

### 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

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### 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

### 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

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### 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

### 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 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

### 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

### 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

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero

### 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

### 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:

### 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

### 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.

### 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

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

### 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,

### 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

### 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:

### 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

### 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

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### 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

### 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

### 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

### 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)

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### 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

### 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

### 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.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### 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)(

### 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

### = [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

### 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

### 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

### 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.

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

### 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

### Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

### LINEAR ALGEBRA. September 23, 2010

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

### 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

### 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

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

### 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

### 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

### 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

### 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

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### 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

### 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

### 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

### 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

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### 4 Solving Systems of Equations by Reducing Matrices

Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example

### 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

### 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

### Numerical Methods Lecture 2 Simultaneous Equations

Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations Matrix operations: Mathcad is designed to be a tool for quick and easy manipulation of matrix forms

### 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

### 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

### 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

### 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 ),

### 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

### 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

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

### 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

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### 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

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### 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