Chapter 3 - Matrices

# Chapter 3 - Matrices - Chapter 3 Matrices Section 3.1...

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Section 3.1 – Matrix Operations Pg 136, Definition – A matrix is a rectangular array of numbers called the entries , or elements of the matrix. Pg 139, Definition – If A is an matrix and B is an matrix, then the product C = AB is an matrix. The entry of the product is computed as follows: Pg 142, Theorem 3.1 – Let A be an matrix, a standard unit vector, and an standard unit vector. Then a. is the ith row of A and b. is the jth column of A. Pg 147, If A is a square matrix a r and s are nonnegative integers, then 1. 2. Pg 149, Definition – The transpose of an matrix A is the matrix A T obtained by interchanging the rows and columns of A. That is, the ith column of A T is the ith row of A for all i. Pg 149, Definition – A square matrix A is symmetric if . Pg 150, A square matrix A is symmetric iff for all i and j. Section 3.2 – Matrix Algebra Pg 152, Theorem 3.2 – Algebraic Properties of Matrix Addition and Scalar Multiplication – Let A, B, and C be matrices of the same size and let c and d be scalars. Then a. (Commutativity) b. (Associativity) c. d. e. (Distributivity) f. (Distributivity) g. h. Pg 156, Theorem 3.3 – Properties of Matrix Multiplication – Let A, B, and C be matrices (whose sizes are such that the indicated operations can be performed) and let k be a scalar. Then a. (Associativity) b. (Left distributivity) c. (Right distributivity) d. e. (Multiplicative identity) Pg 157, Theorem 3.4 – Properties of the Transpose – Let A and B be matrices (whose sizes are such that the indicated operations can be performed) and let k be a scalar. Then a. b. c. d. e. Pg 159, Theorem 3.5 a. If A is a square matrix, then is a symmetric matrix. b. For any matrix A, and are symmetric matrices. Section 3.3 – The Inverse of a Matrix Pg 161, Definition – If A is an matrix, an inverse of A is an matrix A’ with the property that where is the identity matrix. If such an A’ exists, then A is called invertible . Pg 162, Theorem 3.6 – If A is an invertible matrix, then its inverse is unique. Pg 163, Theorem 3.7 – If A is an invertible matrix, then the system of linear equations given by has the unique solution for any b in R n . Pg 163, Theorem 3.8 – If , then is invertible if , in which case

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## This note was uploaded on 11/10/2011 for the course MATH 511 taught by Professor Staff during the Spring '08 term at Washington State University .

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Chapter 3 - Matrices - Chapter 3 Matrices Section 3.1...

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