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Kunze Linear Algebra
· Problem 5.2.12
Determinants
Mathematics — Methods · multiplicative-determinant-functions
Problem
Let
F
F
F
be a field and
D
D
D
a function on
n
×
n
n \times n
n
×
n
matrices over
F
F
F
(with values in
F
F
F
). Suppose
D
(
A
B
)
=
D
(
A
)
D
(
B
)
D(AB) = D(A)D(B)
D
(
A
B
)
=
D
(
A
)
D
(
B
)
for all
A
A
A
,
B
B
B
. Show that either
D
(
A
)
=
0
D(A) = 0
D
(
A
)
=
0
for all
A
A
A
, or
D
(
I
)
=
1
D(I) = 1
D
(
I
)
=
1
. In the latter case show that
D
(
A
)
≠
0
D(A) \neq 0
D
(
A
)
=
0
whenever
A
A
A
is invertible.
Related theory
Characteristic Functions And Moment Generating Functions
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