130 likes | 1.07k Views
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.
E N D
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 = 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.
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.
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.
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.
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.
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.
Homework • p. 75: 3,6,14,18,28,33,38,39,40,42,51 • p. 85: 1,3,5,6,14,22,25,29