How would I solve part c)? Please explain each step in detail and don't post previous answers to this question. 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.

Consider the provided question, According to you we need to solve only part (c). (c) Given, A be an…

For m x n matrices A and B, we write A > B if (A)ij > (B)ij for all 1 < i B and A < B. For a sequence of m × n matrices, we write lim;,0 Bt = B if lim;→∞(Bt)ij = (B)ij for all 1 0 be an n × n matrix. Prove by induction that At > 0 for all t E N. (b) Let A be an n x n matrix. Show that for any positive integer t we have t-1 (I – A) A* = I – A' k=0 (c) Let A be an n × n matrix. Show that if (I – A) is invertible and lim→∞ At = 0 then +00 00 (I – A)-1, Σ Ak k=0 where, by definition, Eo Ak is shorthand for lim→∞i=o Ak. k=0

For m x n matrices A and B, we write A > B if (A)ij > (B)ij for all 1 < i B and A < B. For a sequence of m × n matrices, we write lim;,0 Bt = B if lim;→∞(Bt)ij = (B)ij for all 1 0 be an n × n matrix. Prove by induction that At > 0 for all t E N. (b) Let A be an n x n matrix. Show that for any positive integer t we have t-1 (I – A) A* = I – A' k=0 (c) Let A be an n × n matrix. Show that if (I – A) is invertible and lim→∞ At = 0 then +00 00 (I – A)-1, Σ Ak k=0 where, by definition, Eo Ak is shorthand for lim→∞i=o Ak. k=0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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How would I solve part c)? Please explain each step in detail and don't post previous answers to this question. 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 <j<n. Analogous definitions apply
to A < B, A > B and A < B. For a sequence of m x n matrices, we write lim→∞ Bt = B if lim-(Bt)ij = (B)ij for all
1<i<m and 1 <j< n.
3.
(a) Let A > 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 have
t-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=0
where, by definition, , Ak is shorthand for limt→0∞ E-0 Ak.
k=0
k=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 if
Ar < 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 that
lim 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 strictly
less than 1. If the corresponding Leontief matrix is invertible, is C productive? Explain your answer.

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