6.3
Matrix Algebra
525
6.3
Matrix Algebra
Objectives
1.
Add and Subtract Matrices
2.
Multiply a Matrix by a Constant
3.
Multiply Matrices
4.
Solve Problems Using Matrices
5.
Recognize the Identity Matrix
With certain restrictions, we can add, subtract, and multiply matrices. In fact,
many problems can be solved using the arithmetic of matrices.
To illustrate the addition of matrices, suppose there are 108 police officers
employed at two different locations:
Andy Nelson/The Christian
Science Monitor via Getty Images
Downtown Station
Male
Female
Day shift
21
18
Night shift
12
6
Suburban Station
Male
Female
Day shift
14
12
Night shift
15
10
The employment information about the police officers is contained in two
matrices.
and
The entry 21 in matrix
indicates that 21 male officers work the day shift at the
downtown station. The entry 10 in matrix
indicates that 10 female officers work
the night shift at the suburban station.
To find the city-wide totals, we can add the corresponding entries of matrices
and :
D
S
c
21
18
12
6
d
c
14
12
15
10
d
c
35
30
27
16
d
S
D
S
D
S
c
14
12
15
10
d
D
c
21
18
12
6
d
We interpret the total to mean:
Male Female
35
30
27
16
Day shift
Night shift

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To illustrate how to multiply a matrix by a number, suppose that one-third of
the officers at the downtown station retire. The downtown staff then would
consist of
officers. We can compute
by multiplying each entry of matrix
by .
2
3
D
2
3
c
21
18
12
6
d
c
14
12
8
4
d
2
3
D
2
3
D
2
3
D
526
Chapter 6
Linear Systems
After retirements, the downtown staff will be
Male Female
14
12
8
4
Day shift
Night shift
These examples illustrate two calculations used in the algebra of matrices,
which is the topic of this section. We begin by giving a formal definition of a
matrix and defining when two matrices are equal.
An
matrix
is a rectangular array of
numbers arranged in
rows
and
columns. We say that the matrix is of
size
(or
order
)
.
Matrices often are denoted by letters such as
,
, and
. To denote the
entries in an
matrix
, we use double-subscript notation: The entry in the
first row, third column is
, and the entry in the th row, th column is
. We can
use any of the following notations to denote the
matrix
:
,
,
rows
columns
Two matrices are equal if they are the same size, with the same entries in cor-
responding positions.
If
and
are both
matrices, then
provided that each entry
in matrix
is equal to the corresponding entry
in matrix
.
The following matrices are equal, because they are the same size and corre-
sponding entries are equal.
The following matrices are not equal, because they are not the same size.
The first matrix is
, and the second is
.
1.
Add and Subtract Matrices
We can add matrices of the same size by adding the entries in corresponding
positions.
3
3
2
3
c
1
2
3
1
2
3
d
£
1
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1
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3
1
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2
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0.5
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B
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[
a
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Equality of Matrices
n
⎫
⎪
⎪
⎪
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⎬
⎪
⎪
⎪
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⎪
⎭
m
⎫
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⎬
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⎭
≥
a
11
a
12
a
13
p
a
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n
a
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a
22
a
23
p
a
2
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o
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o
∞
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a
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Matrices

Let
and
be two
matrices. The sum,
, is the
matrix