How would I solve part b)? 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.

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Description: How would I solve part b)? 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 matr

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3. For m x n matrices A and B, we write A > B if (A)¿j 2 (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 × n matrices, we write lim∞ Bt = B if limt→∞(Bt)ij = (B)ij for all 10 be an n x n matrix. Prove by induction that At 2 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 – * k=0

3. For m x n matrices A and B, we write A > B if (A)¿j 2 (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 × n matrices, we write lim∞ Bt = B if limt→∞(Bt)ij = (B)ij for all 10 be an n x n matrix. Prove by induction that At 2 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 – * k=0

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Types of Bias

Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative parameter being estimated. According to the actual definition of bias, it refers to the tendency of a statistic to overestimate or underestimate a parameter.

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How would I solve part b)? 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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