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f(0) = a, f(n + 1)
f(n) = −6 −5
1
0 f(1) = b, n b
a = f(n) = − 6 f(n − 1) − 5 f(n − 2) (−5)n
4 5
5
−1 −1 b
a + (−1)n
4 −1 −5
1
5 (−1)n
(−5)n
( − b − a) +
(b + 5a)
4
4 f(n) = [8] Solve the recurrence relation
f(0) = a, f(n + 1)
f(n) = 16
10 f(1) = b, n f(n) = b
a = f(n) = f(n − 1) + 6 f(n − 2) (−2)n
5 2 −6
−1
3 b
a + 3n
(−2)n
( − b + 3a) +
(b + 2a)
5
5 3n
5 36
12 b
a b
a , 3 × 3 Exercise Set A (distinct roots), November 3 × 3 Exercise Set A (distinct roots)
Linear Algebra, Dave Bayer, November , [1] Find An where A is the matrix 200
A = 2 1 1
122 0
0
0
200
000
n
n
0
2
3
−3
4 −2 +
−1 0 0 +
3 1 1
=
6
2
3
3 −4
2
−5 0 0
622
n λ = 0, 2, 3 An [2] Find An where A is the matrix 122
A = 0 1 0
221 λ = −1, 1, 3 An (−1)n =
2 1 0 −1
121
0 −1 0
3n 00
0 + 0
0 0 0
1 0 +
2
−1 0
1
121
0 −1 0 [3] Find An where A is the matrix 212
A = 0 1 2
012 λ = 0, 2, 3 An 0
0
0
036
1 −1 −2
0n 3n 0
2 −2 + 2n 0
0 1 2
0
0 +
=
3
3
0 −1
1
012
0
0
0 [4] Find An where A is the matrix 111
A = 0 1 2
021 λ = −1, 1, 3 An 0
0
0
2 −1 −1
011
1
3n (−1)n 0
1 −1 +
0
0
0 +
0 1 1
=
2
2
2
0 −1
1
0
0
0
011 [5] Find An where A is the matrix 201
A = 2 1 1
201 , 3 × 3 Exercise Set A (distinct roots), November 1 0 −1
00
0
402
n
0
1
3
00
0 + −2 2 −1 +
6 0 3
=
3
2
6
−2 0
2
00
0
402
n λ = 0, 1, 3 An [6] Find An where A is the matrix 121
A = 1 2 2
002 4 −4
2
0 0 −4
125
n
n
2
3
0
−2
2 −1 +
0 0 −3 +
1 2 5
=
6
2
3
0
0
0...
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This document was uploaded on 03/24/2014 for the course MATH V2010 at Barnard College.
 Spring '14
 DaveBayer
 Linear Algebra, Algebra

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