Cryptography Reference
In-Depth Information
Here are some properties of invertible matrices.
Theorem A.24
(
Properties of Invertible Matrices
)
Let
R
be a commutative ring with identity,
n
∈
N
, and
A
invertible in
M
n
×
n
(
R
)
. Then each of the following holds.
(
A
−
1
)
−
1
=
A.
(a)
(
A
t
)
−
1
=(
A
−
1
)
t
, where
“
t
”
denotes the transpose.
(b)
(
AB
)
−
1
=
B
−
1
A
−
1
.
(c)
In order to provide a formula for the inverse of a given matrix, we need the
following concept.
Definition A.40 (Adjoint)
Let
R
be a commutative ring with identity. If
A
=(
a
i,j
)
∈
M
n
×
n
(
R
)
, then
the matrix
A
a
=(
b
i,j
)
given by
1)
i
+
j
det(
A
j,i
) = cof(
A
j,i
)=
(
1)
i
+
j
det(
A
i,j
)
t
b
i,j
=(
−
−
is called the
adjoint of
A
.
Some properties of adjoints related to inverses, including a formula for the
inverse, are as follows.
Theorem A.25
(
Properties of Adjoints
)
If
R
is a commutative ring with identity and
A
∈
M
n
×
n
(
R
)
, then each of
the following holds.
(a)
AA
a
= det(
A
)
I
n
=
A
a
A
.
(b)
A
is invertible in
M
n
×
n
(
R
)
if and only if
det(
A
)
is a unit in
R
, in which
case
A
−
1
=
A
a
/
det(
A
)
.
For instance, we will need the following in Example 3.2 on page 112.
Example A.12
If
n
= 2, then the inverse of a nonsingular matrix,
A
=
ab
cd
,
is given by
A
−
1
=
,
d
det(
A
)
b
det(
A
)
−
c
det(
A
)
−
a
det(
A
)
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