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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
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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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)
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)
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