2
5
)
1
6
(
13
=
−
=
Δ
,
1
)
2
3
(
21
=
+
−
−
=
Δ
,
0
)
1
1
(
22
=
+
−
=
Δ
,
1
)
3
2
(
23
=
−
−
=
Δ
,
7
)
1
6
(
31
=
+
=
Δ
,
5
)
3
2
(
32
−
=
+
−
=
Δ
,
8
)
9
1
(
33
−
=
−
=
Δ
. The matrix of the cofactors is:
B
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
=
8
5
7
1
0
1
5
5
5
. The transpose of this matrix is: adj
A
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
=
8
1
5
5
0
5
7
1
5
. The
determinant of
A
is:
5
)
5
(
1
)
5
(
3
)
5
(
1
=
−
+
−
.
___________________________________________________________
SC.2 Matrix Operations
When two
m
×
n
matrices
A
and
B
are added, or subtracted, the result is a matrix
C
whose elements are given by:
ij
ij
ij
b
a
c
±
=
for all
i
and
j
(SC.2.1)
In other words the elements of the two matrices are simply added, or subtracted,
element by element.
When a matrix is multiplied, or divided, by a scalar
k
, each element is multiplied,
or divided, by
k
. Two matrices
A
and
B
can be multiplied together to form a product
matrix
C
if and only if the number of columns of
A
is equal to the number of rows of
B
.
Thus, if
A
is an
m
×
s
matrix and
A
is an
s
×
n
matrix, then
C
is an
m
×
n
matrix whose
elements are given by:
kj
ik
s
k
k
ij
b
a
c
∑
=
=
=
1
for
m
i
...,
,
1
=
and
n
j
...,
,
1
=
(SC.2.2)
In words, the element in the
i
th row and
j
th column of
C
is obtained by taking the
i
th row of
A
and the
j
th column of
B
,
each of which contains
s
elements, multiplying them
element by element and adding the products. These operations are illustrated by the
following example.