8.3. MULTILINEAR MAPS AND DETERMINANTS
367
Example 8.3.17.
An element of Mat
n
.
Z
/
has an inverse in Mat
n
.
Q
/
if its
determinant is nonzero. It has an inverse in Mat
n
.
Z
/
if, and only if, its
determinant is
˙
1
.
Permanence of identitities
Example 8.3.18.
For any be an
n
–by–
n
matrix, let
˛.A/
denote the trans
pose of the matrix of cofactors of
A
. I claim that
(a)
det
.˛.A//
D
det
.A/
n
1
, and
(b)
˛.˛.A//
D
det
.A/
n
2
A
.
Both statements are easy to obtain under the additional assumption that
R
is an integral domain and det
.A/
is nonzero. Start with the equation
A˛.A/
D
det
.A/E
, and take determinants to get det
.A/
det
.˛.A//
D
det
.A/
n
. Assuming that
R
is an integral domain and det
.A/
is nonzero,
we can cancel det
.A/
to get the first assertion. Now we have
˛.A/˛.˛.A//
D
det
.˛.A//E
D
det
.A/
n
1
E;
as well as
˛.A/A
D
det
.A/E
. It follows that
˛.A/ ˛.˛.A//
det
.A/
n
2
A
D
0:
Since det
.A/
is assumed to be nonzero,
˛.A/
is invertible in Mat
n
.F /
,
where
F
is the field of fractions of
R
. Multiplying by the inverse of
˛.A/
gives the second assertion.
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 Fall '08
 EVERAGE
 Algebra, Determinant, Invertible matrix, Integral domain, Matn .Z/

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