Cryptography Reference
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is called
the
n
n
identity matrix
, where 1
F
is the identity of
F
.
Another important aspect of matrices that we will need throughout the text
is motivated by the following. Consider the 2
×
×
2 matrix with entries from
F
:
A
=
ab
cd
,
then
ad
bc
is called the
determinant
of
A
, denoted by det(
A
). More generally,
we may define the determinant of any
n
−
×
n
matrix with entries from
F
for any
n
F
is just det(
r
)=
r
. Thus, we have the
definitions for
n
=1
,
2, and we may now give the general definition inductively.
The definition of the determinant of a 3
∈
N
. The determinant of any
r
∈
×
3 matrix,
a
1
,
1
a
1
,
2
a
1
,
3
a
2
,
1
a
2
,
2
a
2
,
3
a
3
,
1
a
3
,
2
a
3
,
3
A
=
×
is defined in terms of the above definition of the determinant of a 2
2 matrix,
namely, det(
A
) is given by
a
1
,
1
det
a
2
,
2
a
2
,
3
a
3
,
2
a
3
,
3
a
1
,
2
det
a
2
,
1
a
2
,
3
a
3
,
1
a
3
,
3
+
a
1
,
3
det
a
2
,
1
a
2
,
2
a
3
,
1
a
3
,
2
,.
−
Therefore, we may inductively define the determinant of any
n
×
n
matrix in
this fashion. Assume that we have defined the determinant of an
n
×
n
matrix.
Then we define the determinant of an (
n
+1)
×
(
n
+ 1) matrix
A
=(
a
i,j
)as
follows. First, we let
A
i,j
denote the
n
n
matrix obtained from
A
by deleting
the
i
th row and
j
th column. Then we define the
minor
of
A
i,j
at position (
i,j
)
to be det(
A
i,j
). The
cofactor
of
A
i,j
is defined to be
×
1)
i
+
j
det(
A
i,j
)
.
cof(
A
i,j
)=(
−
We may now define the determinant of
A
by
det(
A
)=
a
i,
1
cof(
A
i,
1
)+
a
i,
2
cof(
A
i,
2
)+
···
+
a
i,n
+1
cof(
A
i,n
+1
)
.
This is called the
expansion of a determinant by cofactors
along the
i
th row of
A
. Similarly, we may expand along a column of
A
.
det(
A
)=
a
1
,j
cof(
A
1
,j
)+
a
2
,j
cof(
A
2
,j
)+
···
+
a
n
+1
,j
cof(
A
n
+1
,j
)
,
called the
cofactor expansion along the
j
th column of
A
. Both expansions can
be shown to be equal. Hence, a determinant may be viewed as a function that
assigns a real number to an
n
×
n
matrix, and the above gives a method for
finding that number.
If
A
is an
n
n
matrix with entries from
F
, then
A
is said to be
invertible
,
or
nonsingular
if there is a unique matrix denoted by
A
−
1
×
such that
AA
−
1
=
I
n
=
A
−
1
A.
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