By Richard O Hill

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We shall now discuss what is meant by an LU decomposition in such a situation, and we shall also see an easy, but inefficient, method for obtaining them. An alternative approach and rigorous explanation of an efficient method is given in Appendix At. " This is the simplest thing to do for an introductory course. If an efficient method is desired, Appendix A may be substituted for the balance of this section. 5 Gaussian Elimination as a Matrix Factorization '57 Suppose now that row interchanges are performed on a matrix A during Gaussian elimination.

34). 34), we are done. • WARNING Note the reversed order of the inverses and how the proofrequires this order to be reversed. 35) Elementary Matrices We now turn to elementary matrices. They are invertible, and their inverses are easy to determine. We shall see that elementary matrices are "building blocks" for all invertible matrices, a feature that makes them very important. 6) of Section 1 . 1 . This definition is built upon here to define elementary matrices. 36) DEFINITION Let e be an elementary row operation.

Or -7 = - 10 Y3 = 6 60 Chapter 1 Introduction to Linear Equations and Matrices 2 0 23, 6) and finish by solving [ -� -m�J [-��J We obtain (y1, y2, y3) = ( - 1 2 , 4x2 - x3 - 7Xz + 8X3 x1 + or The answer, which you can easily check, is (x1, x2, x3) Exercise == -1230 3x3 = 6 = ( -2, - 1, 2). UX = Y. 5 In Exercises 1 -8, find the L U decomposition for the given matrix. ) 1. 4. 3 . J _ ] �] [-� =� �] 6. [ � -2� ;5 [� �] [-110 ! r-� �AX =rB,9-16,�1 � 4 -7 2. [ _ � - �J 5. =: � ��1A B. 9. A = [! ] . B = [1 �] 10.