LINEAR ALGEBRA
W W L CHEN
c
W W L Chen, 1982, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
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Chapter 2
MATRICES
2.1. Introduction
A rectangular array of numbers of the form
(1)
a
11
. . .
a
1
n
.
.
.
.
.
.
a
m
1
. . .
a
mn
is called an
m
×
n
matrix, with
m
rows and
n
columns. We count rows from the top and columns from
the left. Hence
(
a
i
1
. . .
a
in
)
and
a
1
j
.
.
.
a
mj
represent respectively the
i
th row and the
j
th column of the matrix (1), and
a
ij
represents the entry
in the matrix (1) on the
i
th row and
j
th column.
Example 2.1.1.
Consider the 3
×
4 matrix
2
4
3
−
1
3
1
5
2
−
1
0
7
6
.
Here
(3
1
5
2)
and
3
5
7
Chapter 2 : Matrices
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Linear Algebra
c
W W L Chen, 1982, 2005
represent respectively the 2nd row and the 3rd column of the matrix, and 5 represents the entry in the
matrix on the 2nd row and 3rd column.
We now consider the question of arithmetic involving matrices. First of all, let us study the problem
of addition. A reasonable theory can be derived from the following definition.
Definition.
Suppose that the two matrices
A
=
a
11
. . .
a
1
n
.
.
.
.
.
.
a
m
1
. . .
a
mn
and
B
=
b
11
. . .
b
1
n
.
.
.
.
.
.
b
m
1
. . .
b
mn
both have
m
rows and
n
columns. Then we write
A
+
B
=
a
11
+
b
11
. . .
a
1
n
+
b
1
n
.
.
.
.
.
.
a
m
1
+
b
m
1
. . .
a
mn
+
b
mn
and call this the sum of the two matrices
A
and
B
.
Example 2.1.2.
Suppose that
A
=
2
4
3
−
1
3
1
5
2
−
1
0
7
6
and
B
=
1
2
−
2
7
0
2
4
−
1
−
2
1
3
3
.
Then
A
+
B
=
2 + 1
4 + 2
3
−
2
−
1 + 7
3 + 0
1 + 2
5 + 4
2
−
1
−
1
−
2
0 + 1
7 + 3
6 + 3
=
3
6
1
6
3
3
9
1
−
3
1
10
9
.
Example 2.1.3.
We do not have a definition for “adding” the matrices
2
4
3
−
1
−
1
0
7
6
and
2
4
3
3
1
5
−
1
0
7
.
PROPOSITION 2A.
(MATRIX ADDITION)
Suppose that
A, B, C
are
m
×
n
matrices. Suppose
further that
O
represents the
m
×
n
matrix with all entries zero. Then
(a)
A
+
B
=
B
+
A
;
(b)
A
+ (
B
+
C
) = (
A
+
B
) +
C
;
(c)
A
+
O
=
A
; and
(d) there is an
m
×
n
matrix
A
such that
A
+
A
=
O
.
Proof.
Parts (a)–(c) are easy consequences of ordinary addition, as matrix addition is simply entrywise
addition. For part (d), we can consider the matrix
A
obtained from
A
by multiplying each entry of
A
by
−
1.
The theory of multiplication is rather more complicated, and includes multiplication of a matrix by a
scalar as well as multiplication of two matrices.
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