This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 006 (Lecture 9) Inverse of a Matrix
Deﬁnition 1. The identity matrix of 1 0 0 0 order n is given by the following n × n matrix: 0 ··· 0 1 ··· 0 0 ··· 0 0 ··· 1 • We denote the identity matrix by I . • Notice that IM = M I = M for any square matrix M • In general, if I is the identity of order n, then IM = M and N I = N where M is an n × m matrix and N is a p × n matrix. Deﬁnition 2. Let M be a square matrix of order n (i.e., n × n matrix) and I be the identity matrix of order n. If there exists a matrix M −1 such that M −1 M = M M −1 = I, then M −1 is called the inverse of M. Theorem 1. When M = provided that D = 0. Example 1. Find the inverse of 23 12 . ab cd , then M −1 = 1 D d −b −c a where D = ad − bc, Theorem 2. If (M I ) is transformed by row operation into (I B ), then the resulting matrix B is the inverse of M , that is M −1 = B . 1 Example 2. Find the inverse of each of the following matrices −5 −2 −2 1 0 (a) 2 1 0 1 211 (b) 1 1 0 . 110 Notes 1. Inverse of a matrix does not necessarily exist. It exists if and only if the corresponding system of linear equations has unique solution. 2 ...
View
Full
Document
This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.
 Fall '09
 forgot
 Math

Click to edit the document details