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K(A) 1-1A-1| ||AA| ( 에 ||A|| , if ||A-1|| 1스A < 1. || || + ||||

K(A) 1-1A-1| ||AA| ( 에 ||A|| , if ||A-1|| 1스A < 1. || || + ||||

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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Please solve both parts as soon as possible where k is a multiplication factor

(a) Let I be the n x n identity matrix. Show that ||I = 1 for any induced matrix norm and conclude that K(A) > 1 for any n x n matrix A. (b) Consider solving the linear system Ax = b, where A is an n x n matrix and bE R". Due to inexactness and round-off error in b and A, show that the relative error in x can be bounded as ||A¤|| ||Ab|| |||| к(А) ||A|| ||A|| if ||A-| ||AA|| < 1. - |||| If the relative error in b and A is at most e, how many digits of accuracy do you expect to lose (at worst) in computing the solution x?
Transcribed Image Text:(a) Let I be the n x n identity matrix. Show that ||I = 1 for any induced matrix norm and conclude that K(A) > 1 for any n x n matrix A. (b) Consider solving the linear system Ax = b, where A is an n x n matrix and bE R". Due to inexactness and round-off error in b and A, show that the relative error in x can be bounded as ||A¤|| ||Ab|| |||| к(А) ||A|| ||A|| if ||A-| ||AA|| < 1. - |||| If the relative error in b and A is at most e, how many digits of accuracy do you expect to lose (at worst) in computing the solution x?
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