264
10. Matrices
Defnition 10.1 (matrix)
An
m
×
n
(read ‘
m
by
n
’)
matrix
A
= (
a
i,j
) = (
a
i,j
)
i
=1
,
2
,...,m
j
=1
,
2
,...,n
is a rectangular
array of real numbers containing
m
rows
and
n
columns
A
=
a
1
,
1
a
1
,
2
a
1
,
3
···
a
1
,n
a
2
,
1
a
2
,
2
a
2
,
3
a
2
,n
a
3
,
1
a
3
,
2
a
3
,
3
a
3
,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
a
m,
1
a
m,
2
a
m,
3
a
m,n
The entry in the intersection of the
i
th row and
j
th column is denoted by
a
i,j
and is called the
(
i, j
)
th entry
of the matrix. That is, the
frst index
i
oF
a
i,j
denotes the row
and the
second index
j
oF
a
i,j
denotes the column
.
Example 10.2 (matrices)
(a) The matrix
A
= (
a
i,j
) =
p
2
3
1
−
4
2
1
0
5
P
has
m
= 2 rows and
n
= 4 columns, and hence is a 2
×
4 matrix. Here we have
a
1
,
1
= 2,
a
1
,
2
= 3,
a
1
,
3
= 1,
a
1
,
4
=
−
4, and
a
2
,
1
= 2,
a
2
,
2
= 1,
a
2
,
3
= 0,
a
2
,
4
= 5.
(b) The matrix
B
= (
b
i,j
) =
1
−
2
3
−
2
1
4
7
5
9
−
3
−
4
−
8
has
m
= 4 rows and
n
= 3 columns, and hence is a 4
×
3 matrix.
(c) The
row vector a
= (1
,
4
,
−
5) has
m
= 1 rows and
n
= 3 columns, and hence
is a 1
×
3 matrix.
(d) The
column vector
b
=
1
2
3
has
m
= 3 rows and
n
= 1 columns, and hence is a 3
×
1 matrix.
a