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MAT240 Assignment 10 — Partial Solutions
Arthur Fischer
April 1, 2004
Exercise (#17, p.229).
Let
A,B
∈
M
n
×
n
(
F
)
be such that
AB
=

BA
. Prove that if
n
is odd and
F
is not a ﬁeld of characteristic two, then
A
or
B
is not invertible.
Solution.
Assume as in the statement of the problem that
F
is not a ﬁeld of characteristic two, that
n
is odd, and that
A,B
∈
M
n
×
n
(
F
) are such that
AB
=

BA
. By various theorems and previous
exercises, we have
det(
A
)
·
det(
B
) = det(
AB
) = det(

BA
) = (

1)
n
det(
BA
) =

det(
B
)
·
det(
A
)
.
Since
F
is not a ﬁeld of characteristic two, the only ﬁeldelement which is its own negative is 0, and
therefore det(
A
)det(
B
) = 0. By properties of ﬁelds, this only occurs when either det(
A
) = 0 or
det(
B
) = 0, immediately implying that either
A
or
B
is not invertible.
Exercise (#6, p.238).
Prove that if
M
∈
M
n
×
n
(
F
)
can be written in the form
M
=
±
A B
0
C
²
,
where
A
and
C
are square matrices, then
det(
M
) = det(
A
)
·
det(
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 Fall '09
 Jones
 Linear Algebra, Algebra

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