Answer Guide 7
Math 427K: Unique Number 55160
Wednesday, October 19, 2011
1. The Cayley–Hamilton Theorem in linear algebra says that every matrix satisfies its char
acteristic equation.
This means that if the eigenvalues of
A
are solutions to the scalar
equation
aλ
2
+
bλ
+
c
= 0
,
then
A
itself solves the analogous matrix equation
aA
2
+
bA
+
cI
=
0
0
0
0
.
Do not try to prove this!
Simply verify that it works for the particular matrix
A
=
1
9
5
9
.
Solution:
The characteristic equation of
A
is
0 = det (
A

λI
) =
λ
2

10
λ

36
.
So we compute that
A
2

10
A

36
I
=
1
9
5
9
2

10
1
9
5
9

36
1
0
0
1
=
46
90
50
126

10
90
50
90

36
0
0
36
=
0
0
0
0
.
(
F
)
2. Find the general solution of
X
0
=
LX
if
L
=
5

4
8

7
.
Solution:
The characteristic polynomial of the matrix is 0 =
λ
2
+ 2
λ

3 = (
λ
+ 3)(
λ

1).
An eigenvector associated to the eigenvalue
λ
=

3 is
1
2
; and an eigenvector associated
to the eigenvalue
λ
= 1 is
1
1
.
(Any nonzero multiples work as well.) So the general
solution is
X
(
t
) =
αe

3
t
1
2
+
βe
t
1
1
.
(
F
)
1
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3. The mixing example I gave in the lecture introducing systems led to the system
X
0
=
MX
,
where
M
=
1
100

1
1
1

1
.
Find its general solution.
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 Spring '11
 DELALLAVE
 Differential Equations, Linear Algebra, Algebra, Equations, Scalar, Characteristic polynomial, Eigenvalue, eigenvector and eigenspace, β, 7 l

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