University of Hong Kong
Fall 2011
Math 1813
Mathematical Methods for Actuarial Science
Instructor: Siye Wu
Problem Set 6
(Optional
1
, due at 17:00, Friday, 2 December)
I. Let
A
=
3
1
1
1
1
1
3
1
1
1
1
1
3
1
1
1
1
1
3
1
1
1
1
1
3
.
(a) Show that 7 is an eigenvalue of
A
without using the characteristic polynomial of
A
.
(b) Given that
α
is the other eigenvalue of
A
with multiplicity 4, find
α
and its corresponding
eigenspace.
(c) Write
A
= 2
I
5
+
uu
T
, where
u
= (1
,
1
,
1
,
1
,
1)
T
. Find
A
−
1
by assuming that it is of the form
βI
5
+
γuu
T
for some
β,γ
∈
R
.
II. Let
f
(
x,y
) =
x
4
+
y
4
+ 4
kxy
and
g
(
x,y,z
) =
x
+
1
2
y
2
+
y
+
xz

1
2
z
2

3
2
.
(a) Find the value(s) of
k
so that the surfaces
z
=
f
(
x,y
) and
g
(
x,y,z
) = 0 intersect orthogonally
at the point (1
,
0
,
1)
T
. [Note: Two surfaces intersect orthogonally at a point if their normal vectors
are perpendicular at that point.]
(b) In the above situation(s), find the equation of the plane through (1
,
0
,
1)
T
that is perpendicular
to both surfaces.
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 Fall '11
 Wu
 Math, Actuarial Science, Partial differential equation, wave equation, Siye Wu

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