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Lecture_1_2

# Lecture_1_2 - Lecture Note 1 Sept 11 15 2006 Dr Jeff...

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Unformatted text preview: Lecture Note 1: Sept 11 - 15, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu 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. LU-Factorization LINEAR EQUATIONS AND MATRICES 2 Tutorial Classes : (W9) Wednesday, 4:30-5:15pm, (H9) Thursday, 4:30-5:15 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 non-invertible ). 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 non-invertible 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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Lecture_1_2 - Lecture Note 1 Sept 11 15 2006 Dr Jeff...

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