lecture9 - Math 006 (Lecture 9) Inverse of a Matrix...

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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 ...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture9 - Math 006 (Lecture 9) Inverse of a Matrix...

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