Consider the sequence defined recursively by a1 = -1. 02-2, an+1=-8an-1-9an We can use matrix diagonalization to find an explicit formula for an a. Find a matrix that satisfies M= D= 0 p-1= -9 b. Find the appropriate exponent & such that an - Mk Gn+1 k-n-1 c. Find a diagonal matrix D and an invertible matrix P such that M PDP-¹, -1 3--12 0 d. Find P-1, -8 M- 4/3 0₂ -8 1/6 -1/3 e. Find MPD P-1, an an+1 -1/6 = M an -8 f. Use parts b. and e. to find a d g. You can find an explicit formula for a, using part b. and a formula for MPDP-1. Try to develop this formula. Use your formula to verify the answer for part f
Consider the sequence defined recursively by a1 = -1. 02-2, an+1=-8an-1-9an We can use matrix diagonalization to find an explicit formula for an a. Find a matrix that satisfies M= D= 0 p-1= -9 b. Find the appropriate exponent & such that an - Mk Gn+1 k-n-1 c. Find a diagonal matrix D and an invertible matrix P such that M PDP-¹, -1 3--12 0 d. Find P-1, -8 M- 4/3 0₂ -8 1/6 -1/3 e. Find MPD P-1, an an+1 -1/6 = M an -8 f. Use parts b. and e. to find a d g. You can find an explicit formula for a, using part b. and a formula for MPDP-1. Try to develop this formula. Use your formula to verify the answer for part f
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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Transcribed Image Text:Consider the sequence defined recursively by a1 = -1. 02-2, an+1-8an-1-9an We can use matrix diagonalization to find an explicit formula
for an
a. Find a matrix that satisfies
M=
D=
0
p-1=
-9
b. Find the appropriate exponent & such that
an
- Mk
On+1
k-n-1
c. Find a diagonal matrix D and an invertible matrix P such that M PDP-¹,
-1
0
d. Find P-1,
-8
M
4/3
0₂
-8
1/6
-1/3
e. Find MPD P-1,
an
Gn+1
-1/6
= M
an
P
-8
f. Use parts b. and e. to find ae
d
g. You can find an explicit formula for a, using part b. and a formula for MPDP-1. Try to develop this formula. Use your formula to verify the
answer for part f
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