sider the sequence de a. Find a matrix that satisfies M = D = d. Find P-1 an [an+1] b. Find the appropriate exponent k such that an = Mk a1 α₂ [an+1_ k c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 8-189 P = P-¹ = e. Find M5 = PD5 P-1. 188 f. Use parts b. and e. to find a. M5 = = M [an-1 an 4, an+1 = Tan-1 odn. We can use matrix diagonalization to find explicit mula for Un. 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.
sider the sequence de a. Find a matrix that satisfies M = D = d. Find P-1 an [an+1] b. Find the appropriate exponent k such that an = Mk a1 α₂ [an+1_ k c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-¹. 8-189 P = P-¹ = e. Find M5 = PD5 P-1. 188 f. Use parts b. and e. to find a. M5 = = M [an-1 an 4, an+1 = Tan-1 odn. We can use matrix diagonalization to find explicit mula for Un. 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.
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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Plz complete
![Consider the sequence defined recursively
by a₁ = 2, a₂ = 2, an+1 = 7an-1 — 6an. We can use matrix diagonalization to find an explicit formula for an.
an-1
- M
+]
=
an
a. Find a matrix that satisfies
188
b. Find the appropriate exponent k such that
an
[2₂]
an+1,
M
=
=
D =
P-1
c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-1.
-188
d. Find P-¹.
= Mk
=
M5
=
=
an
e. Find M5 = PD5P-¹.
188
f. Use parts b. and e. to find a6.
an+1.
α6 =
g. You can find an explicit formula for an using part b. and a formula for M¹ = PD¹P-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%2F5a490cc6-ce96-416e-8b1c-a10bd907f477%2F5g6of09_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the sequence defined recursively
by a₁ = 2, a₂ = 2, an+1 = 7an-1 — 6an. We can use matrix diagonalization to find an explicit formula for an.
an-1
- M
+]
=
an
a. Find a matrix that satisfies
188
b. Find the appropriate exponent k such that
an
[2₂]
an+1,
M
=
=
D =
P-1
c. Find a diagonal matrix D and an invertible matrix P such that M = PDP-1.
-188
d. Find P-¹.
= Mk
=
M5
=
=
an
e. Find M5 = PD5P-¹.
188
f. Use parts b. and e. to find a6.
an+1.
α6 =
g. You can find an explicit formula for an using part b. and a formula for M¹ = PD¹P-¹. Try to develop this formula. Use your formula to verify the answer for part f.
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