Consider the sequence defined recursively by a₁ = -1, a2 = 2, an+1 = -4an-1-5an. We can use matrix diagonalization to find an explicit formula for an. an-1 a. Find a matrix that satisfies = M an 188 b. Find the appropriate exponent k such that an = Mk an+1] M = k= c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 1.88 8-18.8 P = D = d. Find P-1 P-¹ = an [an+1] e. Find M5 PD5 P-1. 188 f. Use parts b. and e. to find aç. M5 = a6 = g. You can find an explicit formula for an using part b. and a formula for M* = PDP-1. Try to develop this formula. Use your formula to verify the answer for part f
Consider the sequence defined recursively by a₁ = -1, a2 = 2, an+1 = -4an-1-5an. We can use matrix diagonalization to find an explicit formula for an. an-1 a. Find a matrix that satisfies = M an 188 b. Find the appropriate exponent k such that an = Mk an+1] M = k= c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 1.88 8-18.8 P = D = d. Find P-1 P-¹ = an [an+1] e. Find M5 PD5 P-1. 188 f. Use parts b. and e. to find aç. M5 = a6 = g. You can find an explicit formula for an using part b. and a formula for M* = PDP-1. Try to develop this formula. Use your formula to verify the answer for part f
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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![Consider the sequence defined recursively by a₁ =
[an-1
a. Find a matrix that satisfies
an
M
=
D
b. Find the appropriate exponent k such that
an
[₂]
an+1,
P-1
= Mk
d. Find P-¹.
=
M5
k
c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹.
1.89
-
e. Find M5 = PD5P-¹.
An
an+1_
I
-
M
-1, a2 = 2, an+1 = -4an-1 - 5an. We can use matrix diagonalization to find an explicit formula for an.
f. Use parts b. and e. to find a6.
a6 =
g. You can find an explicit formula for an using part b. and a formula for Mk = PDP-¹. Try to develop this formula. Use your formula to verify the answer for part f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff413a13f-4ccc-4089-b9ed-1c21a38ff098%2F24bd6026-ba03-420b-9fbe-e259357f6c2d%2Fpcd0oh_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the sequence defined recursively by a₁ =
[an-1
a. Find a matrix that satisfies
an
M
=
D
b. Find the appropriate exponent k such that
an
[₂]
an+1,
P-1
= Mk
d. Find P-¹.
=
M5
k
c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹.
1.89
-
e. Find M5 = PD5P-¹.
An
an+1_
I
-
M
-1, a2 = 2, an+1 = -4an-1 - 5an. We can use matrix diagonalization to find an explicit formula for an.
f. Use parts b. and e. to find a6.
a6 =
g. You can find an explicit formula for an using part b. and a formula for Mk = PDP-¹. Try to develop this formula. Use your formula to verify the answer for part f.
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