4. MATRICES
170
4. Matrices
4.1. Definitions.
Definition
4.1.1
.
A
matrix
is a rectangular array of numbers. A matrix with
m
rows and
n
columns is said to have
dimension
m
×
n
and may be represented as
follows:
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
= [
a
ij
]
Definition
4.1.2
.
Matrices
A
and
B
are
equal
,
A
=
B
, if
A
and
B
have the
same dimensions and each entry of
A
is equal to the corresponding entry of
B
.
Discussion
Matrices have many applications in discrete mathematics.
You have probably
encountered them in a precalculus course. We present the basic definitions associated
with matrices and matrix operations here as well as a few additional operations with
which you might not be familiar.
We often use capital letters to represent matrices and enclose the array of numbers
with brackets or parenthesis; e.g.,
A
=
a
b
c
d
or
A
=
a
b
c
d
. We do not use simply
straight lines in place of brackets when writing matrices because the notation
a
b
c
d
has a special meaning in linear algebra.
A
= [
a
ij
] is a shorthand notation often used
when one wishes to specify how the elements are to be represented, where the first
subscript
i
denotes the row number and the subscript
j
denotes the column number
of the entry
a
ij
. Thus, if one writes
a
34
, one is referring to the element in the 3rd
row and 4th column.
This notation, however, does not indicate the dimensions of
the matrix. Using this notation, we can say that two
m
×
n
matrices
A
= [
a
ij
] and
B
= [
b
ij
] are equal if and only if
a
ij
=
b
ij
for all
i
and
j
.
Example
4.1.1
.
The following matrix is a
1
×
3
matrix with
a
11
= 2
,
a
12
= 3
,
and
a
13
=

2
.
h
2
3

2
i
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4. MATRICES
171
Example
4.1.2
.
The following matrix is a
2
×
3
matrix.
0
π

2
2
5
0
4.2. Matrix Arithmetic.
Let
α
be a scalar,
A
= [
a
ij
] and
B
= [
b
ij
] be
m
×
n
matrices, and
C
= [
c
ij
] a
n
×
p
matrix.
(1) Addition:
A
+
B
= [
a
ij
+
b
ij
]
(2) Subtraction:
A

B
= [
a
ij

b
ij
]
(3) Scalar Multiplication:
αA
= [
αa
ij
]
(4) Matrix Multiplication:
AC
=
"
n
X
k
=1
a
ik
c
kj
#
Discussion
Matrices may be added, subtracted, and multiplied, provided their dimensions
satisfy certain restrictions. To add or subtract two matrices, the matrices must have
the same dimensions.
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 Fall '08
 PenelopeKirby
 Matrices, Diagonal matrix, aik ckj

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