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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 12 System of Linear Equations
Objectives Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method
Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]
Matrix Algebra Row 1 Row 3 Column 2Column m n x m Matrix
Matrix Algebra 3 rd Row 2 nd Column
Matrix Algebra 1 Row, m Columns Row Vector
Matrix Algebra n Rows, 1 Column Column Vector
Matrix Algebra If n = m Square Matrix e.g. n=m=5 Main Diagonal
Matrix Algebra Special Types of Square Matrices Symmetric: a ij = a ji
Matrix Algebra Diagonal: a ij = 0, i j Special Types of Square Matrices
Matrix Algebra Identity: a ii =1.0 a ij = 0, i j Special Types of Square Matrices
Matrix Algebra Upper Triangular Special Types of Square Matrices
Matrix Algebra Lower Triangular Special Types of Square Matrices
Matrix Algebra Banded Special Types of Square Matrices
Matrix Operating Rules - Equality [A] mxn =[B] pxq n=pm=qa ij =b ij
Matrix Operating Rules - Addition [C] mxn = [A] mxn +[B] pxq n=p m=q c ij = a ij +b ij
Matrix Operating Rules - Addition Properties [A]+[B] = [B]+[A] [A]+([B]+[C]) = ([A]+[B])+[C]
Multiplication by Scalar
Matrix Multiplication [A] n x m. [B] p x q = [C] n x q m=p
Matrix Multiplication - Properties Associative: [A]([B][C]) = ([A][B])[C] If dimensions suitable Distributive: [A]([B]+[C]) = [A][B]+[A] [C] Attention: [A][B] [B][A]
Operations - Transpose
Operations - Inverse [A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular
Operations - Trace Square Matrix tr[A] = a ii
Linear Equations in Matrix Form
Homework Problems 9.1, 9.2, 9.3 Due Date: Oct 6
4.1 Introduction to Matrices
Matrices A matrix is a rectangular array of quantities (numbers, expressions or function), arranged in m rows and n columns x 3y.
Matrix Definition: An array of numbers in m rows and n colums is called an mxn matrix A square matrix of order n, is an (nxn) matrix.
Mathematics. Matrices and Determinants-1 Session.
Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Three Linear Algebraic Equations.
3_3 An Useful Overview of Matrix Algebra
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Matrix Operations. Matrix Notation Example Equality of Matrices.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods.
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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 15 Solution of Systems of Equations.
ECIV 520 Structural Analysis II Review of Matrix Algebra.
ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW II.
Part 3 Chapter 8 Linear Algebraic Equations and Matrices PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright © The.
Matrices MSU CSE 260.
CE 311 K - Introduction to Computer Methods Daene C. McKinney
Chapter 1: Matrices Definition 1: A matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns. EXAMPLE:
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