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Unformatted text preview: Lecture Note 1: Sept 11  15, 2006 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff ChakFu WONG 1 L INEAR E QUATIONS AND M ATRICES 1. Linear Systems 2. Matrices 3. Dot Product and Matrix Multiplication 4. Properties of Matrix Operations 5. Solutions of Linear Systems of Equations 6. The Inverse of A Matrix 7. LUFactorization LINEAR EQUATIONS AND MATRICES 2 Tutorial Classes : (W9) Wednesday, 4:305:15pm, (H9) Thursday, 4:305:15 pm Place : W9: SC L2, H9: LHC 103. LINEAR EQUATIONS AND MATRICES 3 T HE I NVERSE OF A M ATRIX THE INVERSE OF A MATRIX 4 Definition: An n × n matrix A is called nonsingular (or invertible ) “if" there exists an n × n matrix B such that AB = BA = I n . The matrix B is called the inverse of A . If there exists no such matrix B , then A is called singular (or noninvertible ). Remark From the preceding definition, it follows that AB = BA = I n , then A is also an inverse of B . THE INVERSE OF A MATRIX 5 Remarks Singular (or noninvertible matrix): 1. A matrix A that has no inverse matrix is said to be singular. 2. Any square matrix whose reduced row echelon form is not the identity matrix is singular. THE INVERSE OF A MATRIX 6 Example 1 Let A = 2 3 2 2 and B =  1 3 2 1 1 Since AB = BA = I 2 , we conclude that B is the inverse of A and that A is nonsingular. Theorem 0.1 An inverse of a matrix, if it exists, is unique. Proof : Let B and C be inverse of A . Then BA = AC = I n . Therefore B = BI n = B ( AC ) = ( BA ) C = I n C = C, which completes the proof. THE INVERSE OF A MATRIX 7 Example 2 Let A = 1 2 3 4 To find A 1 , we let A 1 = a b c d Then we must have AA 1 = 1 2 3 4 a b c d = I 2 = 1 0 0 1 THE INVERSE OF A MATRIX 8 so that a + 2 c b + 2 d 3 a + 4 c 3 b + 4 d = 1 0 0 1 Equating corresponding entries of these two matrices, we obtain the linear system a + 2 c = 1 , 3 a + 4 c = 0 , and b + 2 d = 0 , 3 b + 4 d = 1 . The solutions are a = 2 ,c = 3 2 ,b = 1 ,d = 1 2 (Exercise). THE INVERSE OF A MATRIX 9 Moreover, since the matrix a b c d =  2 1 3 2 1 2 also satisfies the property that  2 1 3 2 1 2 1 2 3 4 = 1 0 0 1 we conclude that A is nonsingular and that A 1 =  2 1 3 2 1 2 . THE INVERSE OF A MATRIX 10 Remark Not every matrix has an inverse. For instance, consider the following example . THE INVERSE OF A MATRIX 11 Example 3 Let A = 1 2 2 4 ....
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 Spring '09
 JeffWong
 Linear Algebra, Algebra, Matrix Operations, Department of Mathematics Chinese University of Hong Kong

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