Biomedical Engineering Reference
In-Depth Information
The first of these relations is obtained by setting the indices
p
and
k
equal in
(A.110) and then expanding the determinant. The second is obtained from the first
by setting the indices
n
and
j
equal in first. The third is obtained from the second by
setting the indices
i
and
m
equal in second.
Example A.8.1
Derive the first of (A.111) from (A.110).
Solution: The first of (A.111) is obtained from (A.110) by setting the indices
p
and
k
equal in (A.110) and then expanding the determinant:
d
im
d
in
d
ik
e
ijk
e
mnk
¼
d
jm
d
jn
d
jk
;
d
km
d
kn
3
one finds that
e
ijk
e
mnk
¼
3
d
im
d
jn
d
im
d
jk
d
kn
3
d
in
d
jm
þ d
in
d
km
d
jk
þ d
ik
d
jm
d
kn
d
ik
d
km
d
jn
:
Carrying out the indicated summation over the index
k
in the expression above,
e
ijk
e
mnk
¼
3
d
im
d
jn
d
im
d
jn
3
d
in
d
jm
þ d
in
d
jm
þ d
in
d
jm
d
im
d
jn
:
This is the desired result, the first of (A.111).
□
Example A.8.2
Prove that Det(
AB
)
¼
Det
A
Det
B
.
Solution: Replacing
A
in (A.107) by
C
and selecting the values of
mnp
to be 123,
then (A.107) becomes
Det
C
¼
e
ijk
C
i1
C
j2
C
k3
¼
e
ijk
C
1i
C
2j
C
3k
:
Now
C
is replaced by the product
A
B
using
C
i1
¼
A
im
B
m1
;
C
j2
¼
A
jn
B
n2
;
C
k3
¼
A
kp
B
p3
;
thus
Det
A
B
¼
e
ijk
A
im
B
m1
A
jn
B
n2
A
kp
B
p3
;
or Det
A
B
¼ð
e
ijk
A
im
A
jn
A
kp
Þ
B
m1
B
n2
B
p3
;
where the order of the terms in the second sum has been rearranged from the first.
Comparison of the first four rearranged terms from the second sum with the right-
hand side of (A.107) shows that the first four terms in the sum on the right may
be replaced by
e
mnp
Det
A
; thus applying the first equation of this solution again with
C
replaced by
B
, the desired result is obtained:
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