Biomedical Engineering Reference
In-Depth Information
It is important to be able to construct the inverse of a linear transformation,
r
, if it exists. The inverse transformation exists if
A
1
can be
constructed from
A
, thus the question is one of the construction of the inverse of a
square matrix. The construction of the inverse of a matrix involves the determinant of
the matrix and the matrix of the cofactors. DetA denotes the determinant of
A
.A
matrix is said to be
singular
if its determinant is zero,
non
-
singular
if it is not. The
cofactor of the element
A
ij
of
A
is denoted by co
A
ij
and is equal to (
A
1
r
¼
A
t
,
t
¼
1)
i+j
times the
determinant of a matrix constructed from the matrix
A
by deleting the row and column
in which the element
A
ij
occurs. Co
A
denotes a matrix formed of the cofactors co
A
ij
.
Example A.5.2
2
3
ade
dbf
efc
4
5
.
Compute the matrix of cofactors of
A
;
A
¼
f
2
),
Solution: The cofactors of the distinct elements of the matrix
A
are co
a
¼
(
bc
e
2
), co
c
d
2
), co
d
co
b
¼
(
ac
¼
(
ab
¼
(
dc
fe
), co
e
¼
(
df
eb
), and co
f
¼
(
af
de
); thus the matrix of cofactors of
A
is
2
4
3
5
:
f
2
bc
ð
dc
fe
Þð
df
eb
Þ
e
2
coA
¼
ð
dc
fe
Þ
ac
ð
af
de
Þ
d
2
ð
df
eb
Þð
af
de
Þ
ab
The formula for the inverse of
A
is written in terms of
coA
as
T
Det
A
;
¼
ð
coA
Þ
A
1
(A.46)
T
is the matrix of cofactors transposed. The inverse of a matrix is not
defined if the matrix is singular. For every non-singular square matrix
A
the inverse
of
A
can be constructed, thus
where
ð
coA
Þ
A
1
A
1
A
¼
A
¼
1
:
(A.47)
A
1
It follows then that the inverse of a linear transformation
r
¼
A
t
,
t
¼
r
,
exists if the matrix
A
is non-singular, Det
A
6¼
0.
Example A.5.3
Show that the determinant of a 3-by-3-open product matrix,
a
b
, is zero.
Solution:
2
4
3
5
¼
a
1
b
1
a
1
b
2
a
1
b
3
Det
fa bg¼
Det
a
2
b
1
a
2
b
2
a
2
b
3
a
1
b
1
ð
a
2
b
2
a
3
b
3
a
2
b
3
a
3
b
2
Þ
a
3
b
1
a
3
b
2
a
3
b
3
a
1
b
2
ð
a
2
b
1
a
3
b
3
a
3
b
1
a
2
b
3
Þþ
a
1
b
3
ð
a
2
b
1
a
3
b
2
a
3
b
1
a
2
b
2
Þ¼
0
:
Search WWH ::
Custom Search