Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let A be an invertible stochastic matrix. (a) Show that e¹A−¹ = eT. (This proves that the columns of A-¹ each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e¹ A¯¹, and use what you know about A to substitute an expression for eT. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-1 is not stochastic.

Let A be an invertible stochastic matrix. (a) Show that e¹A−¹ = eT. (This proves that the columns of A-¹ each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e¹ A¯¹, and use what you know about A to substitute an expression for eT. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-1 is not stochastic.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please do fast 

Let A be an invertible stochastic matrix. (a) Show that e A−¹ = e. (This proves that the columns of A-1 each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e²A-¹, and use what you know about A to substitute an expression for e™. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-¹ is not stochastic.
Transcribed Image Text:Let A be an invertible stochastic matrix. (a) Show that e A−¹ = e. (This proves that the columns of A-1 each sum to 1, and hence ||A-¹x||1 = 1 whenever ||x||1 = 1.) Hint: Start with e²A-¹, and use what you know about A to substitute an expression for e™. (b) Nevertheless, it's not true in general that the inverse of an invertible stochastic matrix is stochastic. Give an example of an invertible 2×2 matrix A such that A-¹ is not stochastic.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 6 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS