De Sterck P. º3x+ 4y= 5 Equation 1. The vector space operations are the sum A+ Band the scalar multiple cA. opinions or for particular instruments, software, or resources described in this document is intended or should be. POL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Deﬁnition 1 A m × n matrix A is a rectangular array of numbers with m rows and n columns and written as A = a 11 a 12 ··· a 1n a 21 a 22 ··· a 2n a m1 a m2 ··· a mn a. n be the matrix representing the identity map with respect to B 1 and B 2. Consider an experiment with two factors, where one factor, say, Factor B, is nested within Factor A. A matrix is called a square matrix if the number of rows is equal to the number. Matrices and Determinants is a very important topic in Mathematics. Lecture 10 - Solution via Laplace transform and matrix exponential Lecture 11 - Eigenvectors and diagonalization Lecture 12 - Jordan canonical form Lecture 13 - Linear dynamical systems with inputs and outputs Lecture 14 - Example: Aircraft dynamics Lecture 15 - Symmetric matrices, quadratic forms, matrix norm, and SVD. Lecture 8: AES: The Advanced Encryption Standard Lecture Notes on “Computer and Network Security” by Avi Kak ([email protected] 2) w= b 3 (1. Hammond My email is p. Refer to our notes for a detailed explanation. Hence, AA−1 = A−1A = I A matrix must be square to have an inverse, but not all square matrices have an inverse. 23 4 23 14 427 xyz xyz. Matrix inverse if A is square, and (square) matrix F satisﬁes FA = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn't have an inverse, it's called singular or noninvertible by deﬁnition, A−1A = I; a basic result of linear algebra is that AA−1 = I. Created Date: 4/20/1998 12:25:36 PM. We can also denote the product AB by A B (though few people ever do this10). Factor Analysis Model Model Form Factor Model with m Common Factors X = (X1;:::;Xp)0is a random vector with mean vector and covariance matrix. • Show students the formula for matrix multiplication using W|A. This is a very slight revision of the notes used for Math 19b in the Spring 2009 semester. 2 6 4 1 0 3 7 5 (3) The size of a matrix is deﬁned by n mwhere nis the number of rows and mis the number of columns. Error-correcting codes are used, e. Matrix: A matrix is an ordered rectangular array of numbers or functions. Learn Chapter 3 Matrices of Class 12 free with solutions of all NCERT Questions including Examples and Exercises. What a matrix mostly does is to multiply. Having de ned what matrices are, we want to be able to add them, multiply them by scalars, and multiply them by each other. no Theory This note is about physical quantities R 1,,R n. Notes on linear algebra Darij Grinberg Tuesday 13th December, 2016 at 21:44 These notes are frozen in a (very) unﬁnished state. It is the Combination of linear equation. Identity Matrix An identity matrix, denoted by I, is a square matrix with 1’s along the main diagonal and 0’s everywhere else. A list of these are given in Figure 2. It is an collection of elements which is arranges in rows columns. Ullrich Department of Applied Mathematics University of Waterloo March 20th, 2006 These notes have been funded by. Their product is the identity matrix—which does nothing to a vector, so A 1Ax D x. This chapter summarizes some important results of linear and matrix algebra that are instrumental in deriving many statistical results in subsequent chapters. The results are well below the requirement of ≤50% transmitted light specified in USP <671>. 1 Stiﬀness matrix The stiﬀness matrix [k] from the strain-displacement and constitutive rela-tions, [b] and [d] is given in equation (7). Lecture Notes 1: Matrix Algebra Part A: Vectors and Matrices Peter J. 3 Class Notes. A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or | | | | Compact form the above matrix is represented by [a. It then explains how to add and subtract matrices. The reduced row echelon form of an invertible matrix is the identity matrix rref( A) = In. To understand them, refer to our Matrices PDF Class 12. Finding the Determinant of a 3×3 Matrix – Notes Page 1 of 3 Finding the Determinant of a 3×3 Matrix The determinant of a matrix is a value associated with a matrix. Lecture Notes On Binary Choice Models: Logit and Probit Thomas B. Order of a Matrix: If a matrix has m rows and n columns, then its order is written as m × n. Free PDF download of Class 12 Maths revision notes & short key-notes for Determinants of Chapter 4 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books. 2 trace, determinant and rank jABj= jAjjBj (2a) jA 1j= 1 jAj (2b) jAj= Y evals (2c) Tr[A] = X evals (2d). Lesson 5 2x2 Matrices, Determinants and Inverses 4. 4 p 733 # 43, 47, 49 11/14 Mon Matrices Day 5 : Inconsistent and dependent matrices NOTES #6 HW: Review worksheet for quiz - Solutions. The individual items in a matrix are called its elements or entries. Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other. 3R 2+ R1 C. These notes are for a graduate course in linear algebra. 1 Matrix algebra An mby nmatrix Ais an array of complex numbers Aij for 1 i mand 1 j n. 3 p733 43, 47, 49 11/11 Fri Matrices Day 4: Practicing Row Echelon form - NOTES #5 HW: 11. FçÄ Ã Äã ½ÝÊ¥MãÙ®øA¦ Ù ThirdEdion ,Version3. It is an collection of elements which is arranges in rows columns. Gauss-Jordan Elimination and Solution to Linear Equations33 5. Let A= (a ij) and B= (b ij) be two m n matrices over R. DepartmentofMathemacsandComputerScience VirginiaMilitaryInstute. That is, S is the unique matrix such that v j= Xm i=1 s ijw i for each j. Lecture 8: AES: The Advanced Encryption Standard Lecture Notes on “Computer and Network Security” by Avi Kak ([email protected] Mar 25, 2020 - 1. Factor Analysis Model Model Form Factor Model with m Common Factors X = (X1;:::;Xp)0is a random vector with mean vector and covariance matrix. redistributed. In addition, we will for-mulate some of the basic results dealing with the existence and uniqueness of. A vector is a container in. hk 1 Orthogonal Matrix De nition 1. Let N i ⊆ An 2 k be the set of matrices which are nilpotent of order i, i. Before explaining what a singular value decom-position is, we rst need to de ne the singular values of A. square matrices of a certain size, and whose range is the set of all real (or complex) numbers. These notes for a graduate course in set theory are on their way to be-coming a book. PRESENTED BY: Nikhi jainSalini bhadoriaGunjan dwivedi Shilpi barua Ritu Tomar 2. An MMA-matrix is a matrix all of whose positive integer powers are irreducible M-matrices (see Section 2. Created by T. massage the result into canonical form. Identity Matrix An identity matrix, denoted by I, is a square matrix with 1’s along the main diagonal and 0’s everywhere else. 2 Multiplication of Matrices When you add or subtract matrices, the two matrices that you add or subtract must have the same number of rows and. Its transpose is the gradient of y, denoted ∇y. All matrices that we discuss are over the real numbers. Vandermonde matrix all the top row entries have total degree 0, all the second row entries have total degree 1, and so on. ACE Academy Handwritten Class Notes [PDF] - Mathematics - This is Mathematics study material for GATE / IES / PSUs exam preparation in the form of handwritten notes. Two theorems concerning antisymmetric matrices are particularly useful. Download PDF (262KB) Notes by M. Matrices (Pages 572−573) If m and n are positive integers, an m × n matrix is. Lesson 5 2x2 Matrices, Determinants and Inverses 4. Two matrices are called row equivalent if they have the same row spaces. What a matrix is, how we form it and what is its order; Then we see different types of matrix like Square matrix, Zero matrix, Identity Matrix, Row Matrix, Column Matrix etc. Class 12 Maths Notes Chapter 3 Matrices PDF Download Free. Let A be an m-by-n matrix with matrix elements aij and let B be an n-by-p matrix with matrix elements bij. Inverse matrix (of a square matrix) Given an n × n square matrix A, if there exists a matrix B (necessarily of the same size) such that AB = BA = I n, then the matrix B is called the inverse matrix of A, denoted A−1. Zooming in on an image made up of pixels The convention in these notes will follow that of OpenGL, placing the origin in the lower left corner, with that pixel being at location (0,0). Introduction to Graph Theory Dr. VMware Lifecycle Product Matrix Supported Products, as of May 1, 2020 Dates highlighted in red indicate a product version is within 6 months of End of General Support or End of Technical Guidance. These numbers can be changed for matrices of bigger dimensions. Chapter 4 Matrices. Rotation-of-Coordinate-System Matrix. This is a row-stochastic matrix: the entries in each row form a probability distribution (i. An array is a collection of data items, all of the same type, accessed using a common name. Here are examples of applications addressed in Coding the Matrix. Let A and B be n n matrices. Simplifying the nontrivial element of the nal matrix and recognizing the lensmaker’s equation, n2 n1 R2n1 + n1 n2 R1n1 = n2 n1 n1 (1 R2 1 R1) = 1 flens So for a Gaussian thin lens, we have [r3 θ3] = [1 0 1 f 1][r1 θ1] Notice that if the rays pass into from a medium of refractive index n1 a medium of refractive. It plays an important role in the field of computer science. 215 C H A P T E R 5 Linear Transformations and Matrices In Section 3. Revision Notes on Matrices. Tobias This book is intended as an undergraduate text introducing matrix methods as they relate to engi-neering problems. How it has been developed, and current limitations 3. Solving an equation involving an upper triangular matrix 27 11. tech 1st-year Mathematics notes, you can get the complete Study Material in Single Download Link. The matrix Ais the coefficient matrix of the system, X is the andBis the Writing a Matrix Equation Write the system of linear equations as a matrix equation. uk 1 Index Notation Index notation may seem quite intimidating at rst, but once you get used to it, it will allow us to prove some very tricky vector and matrix identities with very little e ort. But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size) The negative of a matrix is also simple:. microwave engineering pdf notes free download 2019, microwave engineering notes jntu, mwe engineering materials pdf book, mwe pdf lecturer notes,. Thenrootsofthecharacteristic equation,formedbyequating the characteristic polynomial to zero, will deﬁne those values ofλ that make the matrix. Whenever we want to get rid of the matrix Ain some matrix equation, we can “hit” A with its inverse A 1 to make it disappear. Subtraction gives 1 2 (A+AT)−B = C − 1 2 (A−AT). The result of the operation is also a matrix. The inverse of a non-singular. Find the 2 2× matrix X that satisfy the equation AX B= 1 3 2 3 = X Question 24 (***) It is given that A and B are 2 2× matrices that satisfy det 18(AB) = and det 3(B−1) = −. Lecture notes based on J. 7) Zero or Null Matrix. This chapter summarizes some important results of linear and matrix algebra that are instrumental in deriving many statistical results in subsequent chapters. 2 trace, determinant and rank jABj= jAjjBj (2a) jA 1j= 1 jAj (2b) jAj= Y evals (2c) Tr[A] = X evals (2d). of Bellman and Kalman engineers and scientists have found in matrix theory a language for repre-senting and analyzing multivariable systems. In is important to understand these two points but also the differences between them. se ebruaryF 14, 2012 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential eA when A is not diagonalisable. 1) For a matrix, the spectrum is just the collection of eigenvalues, so this formula yields a technique for estimating for the top eigenvalue. Shoulder Joint. We will examine three approaches: by entry, by row, and by column. Linear and Matrix Algebra. RS- Chapter 4 5 4. uk or [email protected] Name this matrix [B]. Preface Here are my online notes for my Linear Algebra course that I teach here at Lamar University. MATRICES: BASIC CONCEPTS A matrix, in general sense, represents a collection of information stored or arranged in an orderly fashion. The header file defines the matrix as a two dimensional vector. Linear Transformations and Matrices. Brent Adams (now emeritus at BYU). Two matrices are called row equivalent if they have the same row spaces. Returning to our example, the transpose of matrix M is MT = 2 66 66 66 4 a d g b e h c f i 3 77 77 77 5 : Note, that the row vectors of the original matrix are now the column vectors of the transpose. Hammond 1 of 80. 1 Basic matrix notation We recall that a matrix is a rectangular array or table of numbers. Whatever A does, A 1 undoes. M ATRIX A matrix is a rectangular array of numbers. The result of the operation is also a matrix. The lectures were distributed to the students before class, then posted on a. Addition of matrices obeys all the formulae that you are familiar with for addition of numbers. Linear and Matrix Algebra. Matrix inversion is discussed,with an introduction of the well known reduction methods. Helical Axis and Angles. 1 Matrices 1. To understand them, refer to our Matrices PDF Class 12. 1 De nitions A matrix is a rectangular array of numbers of the form a11 a12 a13 ··· a1n a21 a22 a23 ··· a2n a31 a32 a33 ··· a3n. uk or [email protected] @a0b @b = @b0a @b = a (6) when a and b are K£1 vectors. If Ais Hermi-tian, that is, if A∗ = A, where the asterisk denotes conjugate transpose, then the eigenvalues of the matrix are real and hence α∗ = α, where the asterisk denotes the conjugate in the case of a complex scalar. Each element in a matrix is called an entry. 5 Class Notes. Linear Transformations and Matrices. Candidates can download Determinants and Matrices Study Materials along with previous year questions and detailed solutions PDF from below mentioned links. In addition, we will for-mulate some of the basic results dealing with the existence and uniqueness of. Name this matrix [A]. The Covariance Matrix Deﬁnition Covariance Matrix from Data Matrix We can calculate the covariance matrix such as S = 1 n X0 cXc where Xc = X 1n x0= CX with x 0= ( x 1;:::; x p) denoting the vector of variable means C = In n 11n10 n denoting a centering matrix Note that the centered matrix Xc has the form Xc = 0 B B B B B @ x11 x 1 x12 x2 x1p. Specify the elementary row operations we wish to perform, one operation at a time. He is admin of official website of Govt Emerson College Multan and its social media accounts. Determine which elementary row operation(s) applied to the first matrix will yield the second matrix. For n,m∈ N, Mn×m = Mn×m(C) denotes the space of all n× mcomplex matrices. number of rows = number of columns. Most of the questions involve calculations. , they are non-negative numbers that sum to 1). Two matrices are equal only if they have the same dimensions and each element of one matrix is equal to the corresponding element in the other matrix. Linear Equations and Matrices In this chapter we introduce matrices via the theory of simultaneous linear equations. Linear algebra has two aspects. 8 Corollary 4: A More Practical Condition (special matrices) If L of the FDM (11) is symmetric or similar to a symmetric matrix, then ˆ(L) 1; for any xand t, is also a su cient condition for stability in the Euclidean norm. 3 Class Notes. Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other. The Factor Analysis model assumes that X = + LF + where L = f‘jkgp m denotes the matrix offactor loadings ‘ jk is the loading of the j-th variable on the k-th common factor. Lecture 10 – Solution via Laplace transform and matrix exponential Lecture 11 – Eigenvectors and diagonalization Lecture 12 – Jordan canonical form Lecture 13 – Linear dynamical systems with inputs and outputs Lecture 14 – Example: Aircraft dynamics Lecture 15 – Symmetric matrices, quadratic forms, matrix norm, and SVD. Determinants. Vandermonde matrix all the top row entries have total degree 0, all the second row entries have total degree 1, and so on. To perform an elementary row operation O on an m n matrix A, calculate the product EA, where. ACE Academy Handwritten Class Notes [PDF] - Mathematics - This is Mathematics study material for GATE / IES / PSUs exam preparation in the form of handwritten notes. M ATRIX A matrix is a rectangular array of numbers. Revision Notes on Matrices & Determinants Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other. This set of notes is fairly self contained but there is enough Algebra type problems (arithmetic and occasionally solving equations) that can show up that not having a good background in Algebra. The (i;j)-coe cient of DCis Xn t=1 d itc tj= Xn t=1 (Xm s=1 a isb st)c tj= Xm s=1 a is(Xn t=1 b stc tj) = Xm s=1 a ise sj. Candidates who are ambitious to qualify the Class 12 with good score can check this article for Notes. Lesson 3 Matrix Multiplication. 4 View / Download Exerci. We will start with one that are somewhat arti cial and then move on to the important class of induced matrix norms. Simulating Movement with Euler Angles. 5 for the de nition of irreducibility). Mathematics Notes for Class 12 chapter 3. We introduce the Dirac spinor ﬁeld ↵(x), an object with four complex components labelled by. n be the matrix representing the identity map with respect to B 1 and B 2. Two matrices are called row equivalent if they have the same row spaces. Add a multiple of one row to another row. In matrix algebra, the inverse of a matrix is that matrix which, when multiplied by the original matrix, gives an identity matrix. The 2 2× matrix C represents a rotation by 90 ° anticlockwise about the origin O,. Suppose a Matrix A has ‘m’ rows and ‘n’ columns the order of Matrix A is denoted by ‘m x n’ read as ‘m by n’. Two matrices are equal only if they have the same dimensions and each element of one matrix is equal to the corresponding element in the other matrix. Introduction to C Programming Arrays Overview. ACE Academy Handwritten Class Notes [PDF] - Mathematics - This is Mathematics study material for GATE / IES / PSUs exam preparation in the form of handwritten notes. Let Aand Bhave the same dimensions. edu) February 5, 2020 11:35am c 2020 Avinash Kak, Purdue University Goals: • To review the overall structure of AES and to focus particularly on the four steps used in each round of AES: (1) byte substitution, (2) shift. Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Create a 24× matrix showing the cost per 9. Preface Here are my online notes for my Linear Algebra course that I teach here at Lamar University. Example Here is a matrix of size 2 3 ("2 by 3"), because it has 2 rows and 3 columns: 10 2 015 The matrix consists of 6 entries or elements. DepartmentofMathemacsandComputerScience VirginiaMilitaryInstute. hk 1 Inverse De nition We use I to represent identity matrices, namely, diagonal matrices where all the elements on the main diagonal are 1. Matrices are a vital area of mathematics for electrical circuits, quantum mechanics, programming, and more! The only way for future Einsteins to become proficient in matrices is by steady, systematic practice with in-depth worksheets like these. Two theorems concerning antisymmetric matrices are particularly useful. The two columns on the right have Virtual Lectures, ScreenCasts or MathCasts that you can see and hear on your computer, smart phone, or tablet device using Flash or YouTube. MATRICES AND LINEAR ALGEBRA (2) Since (A −AT)T = AT −A = −(A −AT), it follows that A −AT is skew-symmetric. ; If two matrices are equal, then how to find its elements. If we have available a data set of such ratings, how can we predict new ratings for (i;j) that we. Name this matrix [A]. 1 Basic matrix notation We recall that a matrix is a rectangular array or table of numbers. Suppose the Request matrix is changed like this. A vector is a container in. These methods work well when the preconditioner is a good approximation for A and when linear equations in the preconditioner can be solved quickly. multiresolutions. 2 we provide a quite thorough and reasonably up-to-date numerical treatment of elliptic partial di erential equations. Create a 24× matrix showing the cost per 9. Four Key Ideas 1. We can also denote the product AB by A B (though few people ever do this10). The use of matrices in computer graphics is widespread. However, we will often reach back to. Name this matrix [A]. Multiply a row by a nonzero constant. • Calculate determinants. Given a photo of a whiteboard taken at an angle, synthesize a perspective-free view of the whiteboard. Markers and Measured. Matrices Day 3: Intro to Row operations #4 HW: 11. The product @y @x x is therefore a generalized matrix-vector multiply, which results in a tensor of shape M 1 M D y. The individual items in a matrix are called its elements or entries. An operation. COM Absolute maximum ratings (Ta=25°C) Parameter S D UR E Y G Unit Forward Current IF 25 25 25 25 25 30 mA Power Dissipation Pd 60 60 60 60 60 65 mW Reverse Voltage VR 5 5 5 5 5 5 V Peak Forward. This method has the advantage of leading in a natural way to the concept of the reduced row-echelon form of a matrix. A square matrix A= [aij] is said to be an lower triangular matrix if aij = 0 for ij). Download this lesson as PDF:-Matrices PDF. Chapter 10 – Isoparametric Elements Learning Objectives • To illustrate by example how to evaluate the stresses at a given point in a plane quadrilateral element using Gaussian quadrature • To evaluate the stiffness matrix of the three-noded bar using Gaussian quadrature and compare the result to that found by explicit evaluation of the. Allows us to solve a simultaneous system. Typical and Atypical Childhood Development Module 1: Birth through 3 Years of Age Matrices Page 4 of 13 Typical Developmental Milestones Atypical Development (Missing or Not Meeting Anticipated Milestones) NOTES 6 MONTHS (Birth to Three Years Old) SOCIAL AND EMOTIONAL Knows familiar faces and begins to know if someone is a stranger. 5) (A−1) =(A )−1 (A : n×n, nonsingular). Two matrices A and B are said to be equal, written A = B, if they have the same dimension and their corresponding elements are equal, i. For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns: [− −]. The dimensions (number of rows and columns) should be same for the matrices involved in the operation. Outline Markers and Measured Orientation Vectors. Every elementary matrix has an inverse, which is also elementary. Finding the Determinant of a 3×3 Matrix – Notes Page 1 of 3 Finding the Determinant of a 3×3 Matrix The determinant of a matrix is a value associated with a matrix. Mar 25, 2020 - 1. We call the individual numbers entriesof the matrix and refer to them by their row and column numbers. Matrices helps students learn how to properly use matrix operations such as switching and adding scalar multiples of rows , and gives practice with the applications of an augmented matrix. edu/ ~ mosulliv April 18, 2013 1 Hermitian Matrices We can write any complex matrix as the sum of it’s real part and imaginary part A= Re(A)+iIm(A), where Re(A);Im(A) are both in M n(R). 1 we defined matrices by systems of linear equations, and in Section 3. Linear Equations and Matrices In this chapter we introduce matrices via the theory of simultaneous linear equations. Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-. Using Cramer’s Rule to Solve Three Equations with Three Unknowns – Notes Page 2 of 4 Now we are ready to look at a couple of examples. Let T: V !V be a linear map, and let A be the matrix representing T with respect to B 1 and let B be the matrix representing T with. AB, which has length c, and BC and AC are the legs, with lengths a. A vector is a container in. These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch. USAGE OF THIS DETECTION ALGORITHM Frequency of check depends on how often a deadlock occurs and how many processes will be affected. The notes form the base text for the course ”MAT-62756 Graph Theory”. We will say that an operation (sometimes called scaling) which multiplies a row of a matrix (or an equation) by a nonzero constant is a row operation of type I. • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by deﬁnition, A−1A =I; a basic result of linear algebra is that AA−1 =I we deﬁne negative powers of A via A−k = A−1 k Matrix Operations 2–12. Fomby Department of Economic SMU March, 2010 Maximum Likelihood Estimation of Logit and Probit Models ¯ ® i i i P P y 0 with probability 1-1 with probability Consequently, if N observations are available, then the likelihood function is N i y i y i L iP i 1 1 1. This PDF file for class 12 Mathematics subject's Matrices topic contains brief and concise notes for easy understanding of topics and quick learning. code, a matrix of two rows and two columns has been created. these can all be found in various sources, including the elementary numerical analysis lecture notes of McDonough [1]. He also, again in the. An m x n matrix A is said to be a square matrix if m = n i. Matrix inverse if A is square, and (square) matrix F satisﬁes FA = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by deﬁnition, A−1A = I; a basic result of linear algebra is that AA−1 = I. for the matrix of coefficients. This is a row-stochastic matrix: the entries in each row form a probability distribution (i. This is a very slight revision of the notes used for Math 19b in the Spring 2009 semester. Take each value of λ in turn, substitute it into Equation (1) and solve the resulting homo-geneous system for x using Gaussian elimination. While all statements below regarding the columns of matrices can also be said of rows, in regression applications we will typically be focusing on the columns. The Frobenius norm kk F: Cm n!R. Do not solve. Matrix inversion. ECON 331 Lecture Notes: Ch 4 and Ch 5 1 Matrix Algebra 1. Extracting a Sub-matrix A portion of a matrix can be extracted and stored in a smaller matrix by specifying the names of both matrices and the rows and columns to extract. 1 Singular values Let Abe an m nmatrix. The Frobenius norm kk F: Cm n!R. 3R2 - R3 14. Read more articles. Core Competencies A summary of C. 1 Matrices and Systems of Equations Objective: In this lesson you learned how to use matrices, Gaussian elimination, and Gauss-Jordan elimination to solve systems of linear equations. Matrices and Determinants. For students wishing to deepen their knowledge of matrix algebra in relation to their study of econometrics, I recommend Matrix Algebra by Abadir and Magnus (2005). Lectures notes on orthogonal matrices (with exercises) 92. Let T: V !V be a linear map, and let A be the matrix representing T with respect to B 1 and let B be the matrix representing T with. n be the matrix representing the identity map with respect to B 1 and B 2. 5 0 84 MATRIX OPERATIONS If two matrices have the same dimensions, you can add or subtract them. Mathematics 1st Year All Chapter Notes | Math FSc Part 1 “Class 11 Mathematics Notes” Mathematics-XI (Punjab Text Book Board, Lahore) These Mathematics-XI FSc Part 1 (1st year) Notes are according to “Punjab Text Book Board, Lahore”. 2 trace, determinant and rank jABj= jAjjBj (2a) jA 1j= 1 jAj (2b) jAj= Y evals (2c) Tr[A] = X evals (2d). 1 Frobenius norm De nition 12. Matrix Inverse35 6. solidstatelogic. 221A Lecture Notes Notes on Tensor Product 1 What is “Tensor”? After discussing the tensor product in the class, I received many questions what it means. (Click on the green letters for solutions. Matrices, Linear Algebra and Linear Programming27 1. Linear Transformations and Matrices. The generalized matrix-vector multipy follows the same algebraic rules as a traditional matrix-vector multiply:. Introduction to Graph Theory Dr. Usually we will just call such a matrix stochastic. But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size) The negative of a matrix is also simple:. 1 Singular values Let Abe an m nmatrix. PDF: Matrices Notes Includes Important Matrices Often Tested! Do you know how to add up two numbers using Matrix Multiplication? Read the above notes to find out how!. 4 p 733 # 43, 47, 49 11/14 Mon Matrices Day 5 : Inconsistent and dependent matrices NOTES #6 HW: Review worksheet for quiz - Solutions. Homogeneous Transformation-combines rotation and translation Definition: ref H loc = homogeneous transformation matrix which defines a location (position and orientation) with respect to a reference frame Sequential Transformations Translate by x, y, z Yaw: Rotate about Z, by (270˚ + q) Pitch: Rotate about Yʼby (a+ 90˚) Roll: Rotate about Z”by t,y. An example relationship matrix. Lecture notes (17 pages) : The matrix methods in geometrical optics can be developed in terms of three basic operators (matrices) for reflection, refraction and translation and can be applied with. n be the matrix representing the identity map with respect to B 1 and B 2. To solve for Xin ABCXD= E,. 1 Thinking about entries of AB Recall that we de–ned the product AB of an m n matrix A with an n p. i Preface The problem sets emphasize mechanically working with vectors and matrices, while the exams tend to emphasize conceptual understanding and synthesis. The syntax is: sub_matrix = matrix ( r1 : r2 , c1 : c2 ) ; where r1 and r2 specify the beginning and ending rows and c1 and c2 specify the beginning and ending columns to be. 2103 13 40 0215 11. We have provided Mathematics 1st Year Study Materials and Lecture Notes for CSE, ECE, EEE, IT, Mech, Civil, ANE, AE, PCE, and all other branches. number of rows = number of columns. The Sµ⌫ are 4⇥4matrices,becausetheµ are 4⇥4 matrices. They originated as handwritten notes in a course at the University of Toronto given by Prof. Matrices are the foundation of linear algebra, which the "language" of most machine learning and analytics algorithms. Boston Matrix Notes Developed by the Boston Consulting Group, a US commercial consulting company, it was adapted in 1992 by =mc to better fit charity fundraising and is now widely used throughout the fundraising world. Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong [email protected] A matrix has an inverse exactly when its determinant is not equal to 0. POWER FLOW ANALYSIS Power flow analysis assumption steady-state balanced single-phase network network may contain hundreds of nodes and branches with impedance X specified in per unit on MVA base Power flow equations bus admittance matrix of node -voltage equation is formulated currents can be expressed in terms of voltages. Rotation Matrices 2. learn Linear Algebra or needing a refresher. For n,m∈ N, Mn×m = Mn×m(C) denotes the space of all n× mcomplex matrices. 5 0 84 MATRIX OPERATIONS If two matrices have the same dimensions, you can add or subtract them. 1 Basics on matrices. SMART NOTES Q&A Matrix storage tubes What did the results show? The results showed that less than 10% of the light in the spectral range of 290 nm to 450 nm passed through the walls of the Matrix ScrewTop amber-colored 0. 1 Simplify, simplify, simplify. Mathematics 490 – Introduction to Topology Winter 2007 What is this? This is a collection of topology notes compiled by Math 490 topology students at the University of Michigan in the Winter 2007 semester. Orthogonal matrices are introduced with examples showing application to many problems. This approach provides the ﬁrst real justiﬁcation for the deﬁnition of matrix. There's even a definition of determinant that defines. Lesson 5 2x2 Matrices, Determinants and Inverses 4. Its core purpose is to create clarity across roles. 3 0 4 1 8 1 3 3 7 2 1 4 7. Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-. 1 As it will turn. Lesson 3 Matrix Multiplication. An important fact about solution sets of homogeneous equations is given in the following theorem: Theorem Any linear combination of solutions of Ax 0 is also a solution of Ax 0. However, we will often reach back to. Madas Created by T. • Explain the general method for solving simultaneous equations. We will examine three approaches: by entry, by row, and by column. edu Linear Regression Models Lecture 11, Slide 4. This chapter summarizes some important results of linear and matrix algebra that are instrumental in deriving many statistical results in subsequent chapters. Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-. • Calculate minors and cofactors. DWRABACK: Only works for linear systems. Linear Equations and Matrices In this chapter we introduce matrices via the theory of simultaneous linear equations. It is for the preparation of Matrices & Determinants. If we view the matrix A as a family of column vectors: A = 2 6 4 A1 A2 ¢¢¢ An 3. Example 1: Given matrices P 25× , Q 53× , R 43× , and S 45× , tell whether each product is defined. It begins with the fundamentals of mathematics of matrices and determinants. Revision Notes on Matrices & Determinants Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other. You probably already know how to do this, but we will de ne these operations anyway. Matrices and Determinants. , aij = bij for all i and j. The Sµ⌫ are 4⇥4matrices,becausetheµ are 4⇥4 matrices. Notes on Jordan Canonical Form Eric Klavins University of Washington 2008 1 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix Jm(λ) having the value. 1)and (cA)ij= cAij: (1. Markers and Measured. Be sure to note that W|A uses a period to represent matrix multiplication. Name this matrix [A]. Block tridiagonal matrix inversion and fast transmission calculations Dan Erik Petersena,*, Hans Henrik B. Linear Algebra plays an important role in the subareas of signal processing, control systems, com-. This approach provides the ﬁrst real justiﬁcation for the deﬁnition of matrix. Stiffness Matrix for a Bar Element Example 9 –Space Truss Problem By applying the boundary conditions the global force-displacement equations are: 5 2 3 1 0. If we have available a data set of such ratings, how can we predict new ratings for (i;j) that we. 1 is beautiful but advanced. To solve for Xin ABCXD= E,. treatment decisions. Special Matrices and Vectors29 3. This is a row-stochastic matrix: the entries in each row form a probability distribution (i. Matrices do not have definite value, but determinants have definite value. PART A: MATRICES A matrix is basically an organized box (or "array") of numbers (or other expressions). Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. Buckingham'spi-theorem HaraldHanche-Olsen [email protected] Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation. Similarly, let. We will start with one that are somewhat arti cial and then move on to the important class of induced matrix norms. 6 we showed that the set of all matrices over a field F may be endowed with certain algebraic properties such as addition and multiplication. 3 View / Download Exercise 1. , c ik = a i1 b 1k + a i2 b 2k + a i3 b 3k + + a in b nk The matrix C = [c ik] m×p is the product of A and B. 12 class Maths Notes Chapter 3 MATRICES free PDF| Quick revision Notes class 12 maths CBSE Revision Notes for CBSE Class 12 Mathematics Matrices Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Lesson 5 2x2 Matrices, Determinants and Inverses 4. M-matrix is a positive stable Z-matrix or, equivalently, a semipositive Z-matrix. Matrices and Determinants - Interview Questions and Answers ♣ This Matrices and Determinants Multiple Choice Questions (MCQs) with Answer and Explanation as well as Notes will certainly help Aspirants to improve their knowledge for various Technical Competitive Examinations :. 1 we defined matrices by systems of linear equations, and in Section 3. All the results contained in these notes also apply to real antisymmetric matrices unless otherwise noted. Mathematics 1st Year All Chapter Notes | Math FSc Part 1 "Class 11 Mathematics Notes" Mathematics-XI (Punjab Text Book Board, Lahore) These Mathematics-XI FSc Part 1 (1st year) Notes are according to "Punjab Text Book Board, Lahore". Two matrices are called row equivalent if they have the same row spaces. Example: 1. Welcome to my math notes site. Note that, since the determinant of thecoeﬁcient matrix is zero, row reduction of the augmented matrix must always lead to at leastone row of zeros. Solving Systems of 3 Equations: by hand and with Gaussian Elimination. 2 Page 2 of 5 WWW. In many universities teachers include this chapter in the syllabus of Linear Algebra. , aij = bij for all i and j. In some instances it is convenient to think of vectors as merely being special cases of matrices. Gauss-Jordan Elimination and Solution to Linear Equations33 5. Symmetric Matrices: A square matrix A is symmetric if A = AT. Introduction to C Programming Arrays Overview. Subtraction gives 1 2 (A+AT)−B = C − 1 2 (A−AT). Free PDF download of Class 12 Maths revision notes & short key-notes for Determinants of Chapter 4 to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books. Vectors and Matrices Notes. It then explains how to add and subtract matrices. These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Matrix inversion. MA8251 Notes ENGINEERING MATHEMATICS 2 Unit 1 Matrix Regulation 2017 For Anna University Free download. The individual items in a matrix are called its elements or entries. Matrix inversion is discussed,with an introduction of the well known reduction methods. That is, S is the unique matrix such that v j= Xm i=1 s ijw i for each j. 12 class Maths Notes Chapter 3 MATRICES free PDF| Quick revision Notes class 12 maths CBSE Revision Notes for CBSE Class 12 Mathematics Matrices Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Math 331 Notes on Matrix Multiplication Since matrix multiplication is complicated, we should expect that there are several ways to view it. Here are examples of applications addressed in Coding the Matrix. Thus if the only problems you do are on the problem sets, you will get really good at row-reducing matrices, but. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices. Proof Suppose that A is an m n matrix and suppose that the vectors x1 and x2 n are solutions of the homogeneous equation Ax 0m. The goal of these lecture is to expose the student to the main concepts and tools of random matrices. Determine which elementary row operation(s) applied to the first matrix will yield the second matrix. 4xExample 1: Use Cramer’s Rule to solve 2x+3y−z=1 +y−3z=11 3x−2y+5z=21. This is possible only when you have the best CBSE Class 12 Maths Notes, study material, and a smart preparation plan. If A is an (mxn) and B an (nxp) matrix (A has the same number of columns as B has rows), then we define the product of AB. For reference, some of the basic tools of matrix algebra, probability, and statistics are reviewed in the Appendix. Convention 1 Multi-column matrices are denoted by boldface uppercase letters: for example, A,B,X. Lecture Notes 1: Matrix Algebra Part A: Vectors and Matrices Peter J. Lecture Notes On Binary Choice Models: Logit and Probit Thomas B. SOLID GEOMETRY (a) Area and perimeter Triangle A = 2 1 base height = 2 1 bh Trapezium A = (sum of two parallel sides) height = 2 1 (a + b) h Circle Area = r2 Circumference = 2 r Sector Area of sector = 360 r2 Length of arc = 2 r Cylinder Curve surface area = 2 rh. Matrix Manipulations: Vectors, Matrices, and Arrays. An important fact about solution sets of homogeneous equations is given in the following theorem: Theorem Any linear combination of solutions of Ax 0 is also a solution of Ax 0. About Author Aamir Rana. Before explaining what a singular value decom-position is, we rst need to de ne the singular values of A. Learning matrices help to solve complex problems related to real life situations in an easy manner. We like to measure them in a consistent system of units, such as the SI system, in which the basic units are the meter, kilogram, second, ampere, and kelvin (m, kg, s, A, K). Book: National Council of Educational Research and Training (NCERT) Class: 12th Class. erations, leading variables, free variables, echelon form, matrix, augmented matrix, Gauss-Jordan reduction, reduced echelon form. For n,m∈ N, Mn×m = Mn×m(C) denotes the space of all n× mcomplex matrices. Notes on Matrix Computation University of Chicago, 2014 Vivak Patel September 7, 2014 1. The Frobenius norm kk F: Cm n!R. Structured Analytic Techniques for Improving Intelligence Analysis Prepared by the US Government March 2009. This is an example of a completed RACI chart. , they are non-negative numbers that sum to 1). These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch. 1 Introduction to Graphics. n be the matrix representing the identity map with respect to B 1 and B 2. 1 View / Download Exercise 1. We introduce the Dirac spinor ﬁeld ↵(x), an object with four complex components labelled by ↵ =1,2,3,4. Matrix Manipulations: Vectors, Matrices, and Arrays. Introductory topics of point-set and algebraic topology are covered in a series of ﬁve chapters. In this chapter, we will typically assume that our matrices contain only numbers. How Excel Handles Matrix Math. Orthogonal matrices are introduced with examples showing application to many problems. MUST-HAVE MATH TOOLS FOR GRADUATE STUDY IN ECONOMICS William Neilson Department of Economics University of Tennessee – Knoxville September 2009. Chapter 1 Linear Algebra 1. Expansion of the determinant results in annth order polynomial in λ, known as the characteristic polynomialofA. NOTES AND FORMULAE SPM MATHEMATICS FORM 1 – 3 NOTES 1. For students wishing to deepen their knowledge of matrix algebra in relation to their study of econometrics, I recommend Matrix Algebra by Abadir and Magnus (2005). 1 Matrices and Determinants Class 9th Mathematics in shape of PDF Exercise 1. A matrix has an inverse exactly when its determinant is not equal to 0. Matrix inversion is discussed,with an introduction of the well known reduction methods. Everything I can find either defines it in terms of a mathematical formula or suggests some of the uses of it. code, a matrix of two rows and two columns has been created. Using Cramer's Rule to Solve Three Equations with Three Unknowns - Notes Page 3 of 4 Example 2: Use Cramer's Rule to solve4x −x+3y−2z=5 −y 3z= 8 2x+2y−5z=7. Notes on Matrix Multiplication and the Transitive Closure Instructor: Sandy Irani An n m matrix over a set S is an array of elements from S with n rows and m columns. Chapter 1 Review of Matrices 1. Notes on Symmetric Matrices Prof. A determinant for a given matrix exists only if it is a square matrix. We will say that an operation (sometimes called scaling) which multiplies a row of a matrix (or an equation) by a nonzero constant is a row operation of type I. Structured Analytic Techniques for Improving Intelligence Analysis Prepared by the US Government March 2009. Chapter 4 Matrices. The major components of Corning Matrigel matrix are laminin (~60%), collagen IV (~30%), entactin (~8%) and heparan sulfate proteoglycan. Notes on Matrix Algebra and Inverse Matrices Inverse of a linear transformation Definition: We call a linear transformation T :RRnn→ invertible (also called nonsingular) if it is both one-to- one (if TT() ()xy= then necessarily xy= ) and onto the codomain (for every vector z R∈ n there is a (unique) xR∈ n such that T()xz= ). The individual items in a matrix are called its elements or entries. series) was originally a modiﬁcation and expansion of notes written by Neal Koblitz at the University of Washington, who generously gave permission to use, modify, and distribute his work. Hip Joint. For example, in Figure 1. To get the (i, k)th element c ik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products i. We de ne A+Bto be the m nmatrix C= (c ij. Check Out Engineering Mathematics 1st-year pdf Notes Download. Currently, only the basics of matrix algebra have been completed (products, triangularity, row operations etc. Stiffness Matrix for a Bar Element Example 9 –Space Truss Problem By applying the boundary conditions the global force-displacement equations are: 5 2 3 1 0. But A 1 might not exist. Any errors you might ﬁnd were almost certainly introduced by these revisions and thus are not the fault of the original author. Two matrices are equal only if they have the same dimensions and each element of one matrix is equal to the corresponding element in the other matrix. Revision Notes on Matrices. For example, I2 2 = " 1 0 0 1 # and I3 3 = 2 6 4 1 0 0 0 1 0 0 0 1 3 7 5 If Ais a square matrix, then AI= IA= A. 3 View / Download Exercise 1. Therefore, we will spend most of the course with the book of Strang [9] learning about matrices and applications of matrix theory. 1 Rank-1 model Consider the problem of modeling a quantity y[i;j] that depends on two indices iand j. Learning matrices help to solve complex problems related to real life situations in an easy manner. The inverse of a matrix is denoted by the superscript “-1”. This is a row-stochastic matrix: the entries in each row form a probability distribution (i. Pitfalls and warnings 25 8. the right matrix. Book: National Council of Educational Research and Training (NCERT) Class: 12th Class. A vector is a container in. Andre Lukas Oxford, 2013 3. In a right triangle, the side opposite the right angle is called the hy- potenuse, and the other two sides are called its legs. Determinants. (Click on the green letters for solutions. 5 for the de nition of irreducibility). SMART NOTES Q&A Matrix storage tubes What did the results show? The results showed that less than 10% of the light in the spectral range of 290 nm to 450 nm passed through the walls of the Matrix ScrewTop amber-colored 0. Euler Angles. Rank of a matrix: The row (column) rank of a matrix S : m× n is the dimension of the linear space spanned by its rows (columns). opinions or for particular instruments, software, or resources described in this document is intended or should be. It is assumed that the reader has already studied matrix algebra or linear algebra, how-ever, these notes are completely self-contained. Chapter 2 Matrices 2. The motivation for considering this relatively simple problem is to illustrate how matrix notation and algebra can be developed and used to consider problems such as. This is possible only when you have the best CBSE Class 12 Maths Notes, study material, and a smart preparation plan. Lecture notes (17 pages) : The matrix methods in geometrical optics can be developed in terms of three basic operators (matrices) for reflection, refraction and translation and can be applied with. Learn Chapter 3 Matrices of Class 12 free with solutions of all NCERT Questions including Examples and Exercises. Lecture notes on linear algebra by David Lerner Department of Mathematics University of Kansas and The students of Math 291 (Fall, 2007) These are notes of a course given in Fall, 2007 to the Honors section of our elementary linear algebra course. 2D Jacobian. Thus, the transpose of an m n matrix will be an n m matrix. Some of the results concerning the matrix logarithm are less well known. The mathematical concept of a matrix refers to a set of numbers, variables or functions ordered in rows and columns. matrix1 <- matrix(c(3, 9, -1, 4, 2, 6), nrow = 2. And then, you can see that if you multiply that by a vector, you'll get the same vector back. Introduction to C Programming Arrays Overview. Vectors and Matrices Notes. PDF: Matrices Notes Includes Important Matrices Often Tested! Do you know how to add up two numbers using Matrix Multiplication? Read the above notes to find out how!. This approach provides the ﬁrst real justiﬁcation for the deﬁnition of matrix. 2 trace, determinant and rank jABj= jAjjBj (2a) jA 1j= 1 jAj (2b) jAj= Y evals (2c) Tr[A] = X evals (2d). The determinant of the matrix can be. 1 As it will turn. An array is a collection of data items, all of the same type, accessed using a common name. If the identity matrix is denoted by I and the matrix M is M = 1 2 4 7 8 6 , use the appropriate identity matrix to calculate the following matrix products. Read more articles. A square matrix A= [aij] is said to be an upper triangular matrix if aij = 0 for i>j. FçÄ Ã Äã ½ÝÊ¥MãÙ®øA¦ Ù ThirdEdion ,Version3. Get here NCERT Solutions for Class 12 Maths Chapter 3. This topic covers: - Adding & subtracting matrices - Multiplying matrices by scalars - Multiplying matrices - Representing & solving linear systems with matrices - Matrix inverses - Matrix determinants - Matrices as transformations - Matrices applications. 0 mL storage tubes. For example, in Figure 1. Matrix arithmetic: multiplication 21 7. LINEAR SECOND ORDER ODES. learn Linear Algebra or needing a refresher. Allows us to test for the existance of solutions to simultaneous systems. Two theorems concerning antisymmetric matrices are particularly useful. a) Write down the matrices A and B. He also, again in the. The product of two matrices can also be deﬁned if the two matrices have appropriate dimensions. 3 p733 43, 47, 49 11/11 Fri Matrices Day 4: Practicing Row Echelon form - NOTES #5 HW: 11. Its core purpose is to create clarity across roles. 1 View / Download Exercise 1. Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. This PDF file for class 12 Mathematics subject's Matrices topic contains brief and concise notes for easy understanding of topics and quick learning. For Example, are all zero matrices of the order 1, 2 and 3 respectively. If L is a symmetric matrix then the Eucledian norm jjL jj 2 = p ˆ(L LT) = ˆ(L). 78 KiB, 5283 downloads) Download PDF (1. • Calculate minors and cofactors. Matrices are a vital area of mathematics for electrical circuits, quantum mechanics, programming, and more! The only way for future Einsteins to become proficient in matrices is by steady, systematic practice with in-depth worksheets like these. Here you can find the JNTUA Notes Pdf, Lecture Study Materials & Books related to Engineering departments like ECE, CSE, MECH, EEE and CIVIL branches of 1-1, 1-2, 2-1. Similarly, we want to have measures for how \big" matrices are. COM , [email protected] The motivation for considering this relatively simple problem is to illustrate how matrix notation and algebra can be developed and used to consider problems such as. This is possible only when you have the best CBSE Class 12 Maths Notes, study material, and a smart preparation plan. • Define and form the inverse matrix. As with most. Let T: V !V be a linear map, and let A be the matrix representing T with respect to B 1 and let B be the matrix representing T with. Helical Axis and Angles. Matrices Class 12 Formulas and Notes Class 12 Maths Chapter 3 Matrices Class 12 Formulas & Notes – PDF Download A matrix is a two-dimensional array of numbers, having a fixed number of rows and columns, and containing a number at the intersection of each row and each column.

y3yfnkcr8y18y m9smcs8ack 962y29g3sn6 ghndqk15jt z4pvb14xxdca xmewb23861oun82 88fcfmcqz0gvo ua1mkg3tbpoj1 qtj3ngpslgmm8a0 dtsrptojgkt7 p6tmv0uyob in1ah4djfc cctwbzdvma 9n7am24cxokln2y 3om4zvwcxcoe64p a9w6qpulbh9 6g7by2hnlwbyop lv4h4b4ary5vt 81jtijz76nyoc 484drhgx4h8xy6c qh8t55eyvtqt2t gmlj48ex4nezp ghdyxatidpeqsg 1stfu4ulyxijdl czuogxlkx4 qmxxnw09xzna ujj11fxqj0cj f2ckd7m2zhh6t