Biology Reference
In-Depth Information
Likewise, we could write the system of m linear equations in n
unknowns
e
1
,
e
2
,
,
e
n
:
...
a
11
e
1
þ
a
12
e
2
þ
...
þ
a
1n
e
n
¼
b
1
a
21
e
1
þ
a
22
e
2
þ
...
þ
a
2n
e
n
¼
b
2
(8-61)
.
a
n1
e
1
þ
a
n2
e
þ
...
þ
a
nn
e
¼
b
n
2
n
as the matrix equation:
0
@
1
A
0
@
1
A
¼
0
@
1
A
:
a
11
a
12
...
a
1n
e
b
1
b
2
.
b
m
1
e
a
21
a
22
...
a
2n
2
.
e
n
.
a
m1
a
m2
a
mn
...
This is often written in the more compact form:
A
e ¼
b
:
(8-62)
We say that Eq. (8-62) is the matrix form of the system of Eq. (8-61).
If one examines what we have done, a requirement for the dimensions in
the matrices appears, namely, an m
n matrix multiplying an n
1
matrix gives an m
1 matrix.
Everything we have done is a special (but very important) case of the
following rules governing multiplication of matrices:
(i) If A is an m
n matrix and B is an n
k matrix, then AB is an
m
k matrix; and
(ii) The entry in the i-th row and j-th column of the matrix AB is:
0
@
1
A
¼
b
1j
b
2j
.
b
nj
ð
;
;
...
;
Þ
þ
þ
:
a
i1
a
i2
a
in
a
i1
b
1j
a
i2
b
2j
a
in
b
nj
How does this help us solve a system of linear equations? Actually, we
need to do one more thing before we can accomplish this. Assume
that in the matrix Eq. (8-62), we know the entries of A and b and
want to find the entries of
. If there were a matrix A
1
e
for which
A
1
A
, then multiplying both sides of Eq. (8-62) by A
1
would give
the solution
e ¼ e
A
1
A
A
1
b
e ¼
e ¼
:
