Databases Reference
In-Depth Information
In order to find the inverse of a matrix, we need the concepts of determinant and cofactor.
Associated with each square matrix is a scalar value called the
determinant
of the matrix. The
determinant of a matrix
A
is denoted as
|
|
A
. To see how to obtain the determinant of an
N
×
N
×
×
matrix, we start with a 2
2 matrix. The determinant of a 2
2 matrix is given as
=
a
00
a
01
a
10
a
11
|
A
| =
a
00
a
11
−
a
01
a
10
(B.14)
Finding the determinant of a 2
2 matrix is easy. To explain how to get the determinants of
larger matrices, we need to define some terms.
The
minor
of an element
a
ij
of an
N
×
×
N
matrix is defined to be the determinant of the
N
−
1
×
N
−
1 matrix obtained by deleting the row and column containing
a
ij
. For example,
if
A
is a 4
×
4matrix
⎡
⎤
a
00
a
01
a
02
a
03
a
10
a
11
a
12
a
13
a
20
a
21
a
22
a
23
a
30
a
31
a
32
a
33
⎣
⎦
A
=
(B.15)
then the minor of the element
a
12
, denoted by
M
12
, is the determinant
a
00
a
01
a
03
a
20
a
21
a
23
a
30
a
31
a
33
M
12
=
(B.16)
The cofactor of
a
ij
, denoted by
A
ij
, is given by
i
+
j
M
ij
A
ij
=
(
−
1
)
(B.17)
Armed with these definitions we can write an expression for the determinant of an
N
×
N
matrix as
N
−
1
|
| =
(B.18)
A
a
ij
A
ij
i
=
0
or
N
−
1
|
A
| =
a
ij
A
ij
(B.19)
j
=
0
where the
a
ij
is taken from a single row or a single column. If the matrix has a particular row
or column that has a large number of zeros in it, we would need fewer computations if we
picked that particular row or column.
Equations (
B.18
) and (
B.19
) express the determinant of an
N
×
N
matrix in terms of
determinants of
N
−
1
×
N
−
1 matrices. We can express each of the
N
−
1
×
N
−
1
determinants in terms of
N
−
2
×
N
−
2 determinants, continuing in this fashion until we have
everything expressed in terms of 2
2 determinants, which can be evaluated using (
B.14
).
Now that we know how to compute a determinant, we need one more definition before
we can define the inverse of a matrix. The
adjoint
of a matrix
A
, denoted by (
A
), is a matrix
×
