Problem 2 . Let M(d) be the set of all matrices with the size d x d, where d E N and d≥ 2. (a) Assume that A, B = M(d) are commutative i.e. AB = BA. Then, show that e(A+B) = e¹e³. (b) Use part (a) to show that for a given matrix A E M(d), the matrix eis invertible. (c) Given A = M(d), consider the linear differential equation x'(t) = Ax(t). Let Þ(t) be a fundamental matrix subjected to equation (1). Show that et = Þ(t)Þ−¹(0). (d) Use part (a) and (c) to show that for any t, s ER we have Þ(t)-¹(0)Þ(s)-¹(0) = Þ(t + s)þ−¹(0). (1)
Problem 2 . Let M(d) be the set of all matrices with the size d x d, where d E N and d≥ 2. (a) Assume that A, B = M(d) are commutative i.e. AB = BA. Then, show that e(A+B) = e¹e³. (b) Use part (a) to show that for a given matrix A E M(d), the matrix eis invertible. (c) Given A = M(d), consider the linear differential equation x'(t) = Ax(t). Let Þ(t) be a fundamental matrix subjected to equation (1). Show that et = Þ(t)Þ−¹(0). (d) Use part (a) and (c) to show that for any t, s ER we have Þ(t)-¹(0)Þ(s)-¹(0) = Þ(t + s)þ−¹(0). (1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Problem 2
Let M (d) be the set of all matrices with the size d × d, where d E N and d≥ 2.
(a) Assume that A, B = M(d) are commutative i.e. AB = BA. Then, show that e(A+B) = eªeB¸
(b) Use part (a) to show that for a given matrix A E M(d), the matrix eª is invertible.
(c) Given A E M(d), consider the linear differential equation
x' (t) = Ax(t).
Let Þ(t) be a fundamental matrix subjected to equation (1). Show that etª = Þ(t)Þ−¹(0).
(d) Use part (a) and (c) to show that for any t, s ER we have
Þ(t)Þ¯¹(0)Þ(s)Þ¯¹ (0) = Þ(t + s)Þ¯¹(0).
●
(1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F293ec77a-bb0f-4498-a317-90c98547c742%2F69c41d5c-0fca-47d9-b804-efa5a7764dff%2Fzjdibu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2
Let M (d) be the set of all matrices with the size d × d, where d E N and d≥ 2.
(a) Assume that A, B = M(d) are commutative i.e. AB = BA. Then, show that e(A+B) = eªeB¸
(b) Use part (a) to show that for a given matrix A E M(d), the matrix eª is invertible.
(c) Given A E M(d), consider the linear differential equation
x' (t) = Ax(t).
Let Þ(t) be a fundamental matrix subjected to equation (1). Show that etª = Þ(t)Þ−¹(0).
(d) Use part (a) and (c) to show that for any t, s ER we have
Þ(t)Þ¯¹(0)Þ(s)Þ¯¹ (0) = Þ(t + s)Þ¯¹(0).
●
(1)
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 68 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)