Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 2 - Matrices

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 2 - Matrices

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46 2 Matrices 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5 Applications of Matrix Operations CHAPTER OBJECTIVES Write a system of linear equations represented by a matrix, as well as write the matrix form of a system of linear equations. Write and solve a system of linear equations in the form Use properties of matrix operations to solve matrix equations. Find the transpose of a matrix, the inverse of a matrix, and the inverse of a matrix product (if they exist). Factor a matrix into a product of elementary matrices, and determine when they are invertible. Find and use the -factorization of a matrix to solve a system of linear equations. Use a stochastic matrix to measure consumer preference. Use matrix multiplication to encode and decode messages. Use matrix algebra to analyze economic systems (Leontief input-output models). Use the method of least squares to find the least squares regression line for a set of data. LU A x ± b . Operations with Matrices In Section 1.2 you used matrices to solve systems of linear equations. Matrices, however, can be used to do much more than that. There is a rich mathematical theory of matrices, and its applications are numerous. This section and the next introduce some fundamentals of matrix theory. It is standard mathematical convention to represent matrices in any one of the following three ways. 1. A matrix can be denoted by an uppercase letter such as 2. A matrix can be denoted by a representative element enclosed in brackets, such as ± a ij ² , ± b ² , ± c ² , . . . . A , B , C , . . . . 2.1
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Section 2.1 Operations with Matrices 47 3. A matrix can be denoted by a rectangular array of numbers As mentioned in Chapter 1, the matrices in this text are primarily real matrices. That is, their entries contain real numbers. Two matrices are said to be equal if their corresponding entries are equal. Consider the four matrices and Matrices and are not equal because they are of different sizes. Similarly, and are not equal. Matrices and are equal if and only if REMARK : The phrase “if and only if” means the statement is true in both directions. For example, “ if and only if ” means that implies and implies A matrix that has only one column, such as matrix in Example 1, is called a column matrix or column vector. Similarly, a matrix that has only one row, such as matrix in Example 1, is called a row matrix or row vector. Boldface lowercase letters are often used to designate column matrices and row matrices. For instance, matrix in Example 1 can be partitioned into the two column matrices and as follows. A ± ± 1 3 2 4 ² ± ± 1 3 ± ± 2 4 ² ± ³ a 1 ± a 2 ´ a 2 ± ± 2 4 ² , a 1 ± ± 1 3 ² A C B p . q q p q p x ± 3. D A C B B A D ± ± 1 x 2 4 ² . C ± ³ 13 ´ , B ± ± 1 3 ² , A ± ± 1 3 2 4 ² , EXAMPLE 1 Equality of Matrices ± a 11 a 21 a 31 .
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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 2 - Matrices

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