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(b) Let A be an n x n matrix. Show that for any positive integer t we have t-1 ( 1-A)ΣΑ* =I -Α' k=0

(b) Let A be an n x n matrix. Show that for any positive integer t we have t-1 ( 1-A)ΣΑ* =I -Α' k=0

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 29E
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Need help with part b). 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 <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.
Transcribed Image Text: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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