(a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that
(a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that
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
![(a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show
that
(b) Let
X =
B
I3
I1
X2 A
c(I+tM)¹ c¹(I - tM)
2
-2
1
1
5 2
-1 2
=
9
e³t
-].
3t
f(t) =
and Xo
=
(i) Show that A has a single eigenvalue X. Find A.
(ii) Show that A = AI + N where N is a nilpotent matrix. Use this to compute the
matrix exponential e¹A
(iii) Use the matrix exponential and the method of variation of parameters to solve
x' = Ax+ f(t), x(0) = xo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2Fa827498d-5090-41b6-a161-54f566b853fa%2Fpjnvtio_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show
that
(b) Let
X =
B
I3
I1
X2 A
c(I+tM)¹ c¹(I - tM)
2
-2
1
1
5 2
-1 2
=
9
e³t
-].
3t
f(t) =
and Xo
=
(i) Show that A has a single eigenvalue X. Find A.
(ii) Show that A = AI + N where N is a nilpotent matrix. Use this to compute the
matrix exponential e¹A
(iii) Use the matrix exponential and the method of variation of parameters to solve
x' = Ax+ f(t), x(0) = xo.
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