1 / 8

Properties of Inverse Matrices

Properties of Inverse Matrices. Mark Ginn Math 2240 Appalachian State University. Definition. Last time we said the the inverse of an n by n matrix A is an n by n matrix B where, AB = BA = I n.

rollin
Download Presentation

Properties of Inverse Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Properties of Inverse Matrices Mark Ginn Math 2240 Appalachian State University

  2. Definition • Last time we said the the inverse of an n by n matrix A is an n by n matrix B where, AB = BA = In. • We also talked about how to find the inverse of a matrix and said that not all matrices have inverses (some are singular) so won’t review that here.

  3. Properties of Inverses 1. If A is an invertible matrix then its inverse is unique. 2. (A-1)-1 = A. 3. (Ak)-1= (A-1)k (we will denote this as A-k) 4. (cA)-1 = (1/c)A-1, c ≠ 0. 5. ( AT)-1 = (A-1)T.

  4. Some theorems involving Inverses 1. If A and B are invertible matrices then, (AB)-1 = B-1A-1. 2. If C is an invertible matrix then the following properties hold. a) If AC = BC then A = B. b) If CA = CB then A = B. 3. If A is an invertible matrix, then the system of equations Ax = b has a unique solution given by x = A-1b.

  5. Elementary Matrices • An n by n matrix is called an elementary matrix if it can be obtained from Inby a single elementary row operation. • These matrices allow us to do row operations with matrix multiplication.

  6. Representing Elementary Row Operations Theorem: Let E be the elementary matrix obtained by performing an elementary row operation on In. If that same row operation is performed on an m by n matrix A, then the resulting matrix is given by the product EA.

  7. Row equivalent matrices • Let A and B be m by n matrices. Matrix B is row equivalent to A if there exists a finite number of elementary matrices E1, E2, ... Eksuch that B = EkEk-1 . . . E2E1A.

  8. Homework • p. 75: 3,6,14,18,28,33,38,39,40,42,51 • p. 85: 1,3,5,6,14,22,25,29

More Related