Need help with part d). Please explain each step. Thank you :) For m x n matrices A and B, we write A > B if (A);j > (B)ij for all 1 < i< m and 1 B and A < B. For a sequence of m x n matrices, we write lim→∞ Bt = B if lim-(Bt)ij = (B)ij for all1 O be an n x n matrix. Prove by induction that At > O for all t e N.(b) Let A be an n x n matrix. Show that for any positive integer t we havet-1(I – A) A* = I – A'-k=0(c) Let A be an n x n matrix. Show that if (I – A) is invertible and limt At = O then(I – A)-1 = A*k=0where, by definition, , Ak is shorthand for limt→0∞ E-0 Ak.k=0k=0(d) Let A e (0,1), let r e R" be given by ri = 1 for all1 < i< n, and let A be an n x n matrix. Prove by induction that ifAr < Ar then Aťr < \*r for all t e N.(e) Let A > 0 be an n x n matrix, and suppose the row sums of A are all less than or equal to A E (0, 1). Show thatlim At = O. (Hint: Remind yourself what the sandwich theorem from calculus says.)(f) Let C > O be the consumption matrix of an open economy with n industries, and suppose the row sums of C are all strictlyless than 1. If the corresponding Leontief matrix is invertible, is C productive? Explain your answer.

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Answered: (d) Let A e (0, 1), let r e R" be given…

(d) Let A e (0, 1), let r e R" be given by r; = 1 for all 1

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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