Qualifying Examination
Name:
Math 554
January, 1998
Answering any question you can use the answers to preceding ones
NOTATION.
M
n
(
F
)istheseto
f
n
×
n
matrices
with elements in a
ﬁeld
F
.
1.
For the matrix
A
=
110
0

1

100

2

221
11

10
∈
M
4
(
R
) ﬁnd:
a) The rational form
R
and Jordan canonical form
J
.
[10]
b) An invertible matrix
S
∈
M
4
(
R
) such that
S

1
AS
=
J
.[
5
]
2.
For
A
=(
a
ij
)
∈
M
n
(
F
) with
n
≥
3, let
A
†
a
†
ij
)
∈
M
n
(
F
) be the matrix in
which
a
†
ij
is the cofactor
A
ij
of
a
ij
. Prove that
A
††
= det(
A
)
n

2
A
.
[10]
3.
For all
A, B
∈
M
n
(
F
), set
h
A,B
i
=tr(
AB
).
1. Prove that
h
,
i
is a nondegenerate symmetric bilinear form on
M
n
(
F
). [5]
Fix
C
∈
M
n
(
F
)andset
S
=
{
A
∈
M
n
(
F
):
AC
=
CA
}
.
2. Show that
S
⊥
=
{
BC

CB
:
B
∈
M
}
.
[10]
T
is a
linear operator
on a nonzero
ﬁnite dimensional vector
space
V
over
F
.
4.
The matrix of
T
in some basis of
V
is equal to
λ
000
1
λ
00
01
λ
0
μ
.
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This document was uploaded on 01/25/2012.
 Spring '09
 Matrices

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