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Unformatted text preview: 1 CC2053: Algebra 3 Introduction to Calculus and Linear Algebra 2011/2012 Semester 1 LINEAR ALGEBRA: Matrix Inverse Methods 2 CC2053: Algebra 3 Inverse of a Square Matrix • Let In be an n x n matrix with 1 at the diagonal and zero elsewhere, then In is called the n x n identity matrix • For n = 1,2,3, we have • If M is a square matrix of order n and I is the identity matrix of order n , then I M = M I = M • If A is any m x n matrix and B is any n x p matrix, and I is the identity matrix of order n then A I = A and I B = B = 1 1 2 I = 1 1 1 3 I [] 1 2 = I 3 CC2053: Algebra 3 Example 21: a. b. c. d. i h g f e d c b a 1 1 1 = i h g f e d c b a 1 1 1 i h g f e d c b a i h g f e d c b a f e d c b a 1 1 f e d c b a 1 1 1 f e d c b a f e d c b a = = = 4 CC2053: Algebra 3 Inverse of a Square Matrix Let A be an n x n square matrix. If B is an n x n matrix such that A·B= I n , then B is called the inverse of A , denoted by A1. In that case, we say A is nonsingular or invertible . If A does not have an inverse, then A is said to be singular or noninvertible . I AA AB = = 1 5 CC2053: Algebra 3 Inverse of a Square Matrix • If A·A1 = I , then A1 · A = I . • Inverse of a matrix is unique if it exists. Proof : Let B and C be the inverse of the matrix A, then BA = I and AC = I . Hence, we have C = I C = BAC = B I = B. • (A1 )1 = A. • (A·B)1 = B1 A1 . Proof: Since ABB1 A1 = A I A1 = AA1 = I , thus B1 A1 is the inverse of AB. 6 CC2053: Algebra 3 Example 22: Find M – 1 , where M = 4 3 2 1 7 CC2053: Algebra 3 Example 22: solution Let M 1 = . MM1 = 4 3 2 1 = 1 1 Then Thus, and s r q p s r q p 4 3 1 2 = + = + r p r p 1 4 3 2 = + = + s q s q Rewrite the above systems in augmented matrices 4 3 1 2 1 1 4 3 2 1 Combining the augmented matrices 1 4 3 1 2 1 Perform row operation on the above matrix to obtain s r q p 1 1 and 8 CC2053: Algebra 3 Example 22: solution 1 4 3 1 2 1  1 3 2 1 2 1 2 2 2 1 R R → ~ 2 2 1 3 R R R → Thus M1 = ~  2 1 2 3 1 1 2 1 1 2 1 2 R R R →  2 1 2 3 1 1 2 1 ~  2 1 2 3 1 2 9 CC2053: Algebra 3 Inverse of a Square Matrix M If [ M  I ] is transformed by row operations into [ I  B ], then the resulting matrix...
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This note was uploaded on 03/19/2012 for the course COMP 3868 taught by Professor Keithchan during the Spring '97 term at Hong Kong Polytechnic University.
 Spring '97
 KEITHCHAN

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