46
2
Matrices
2.1
Operations with
Matrices
2.2
Properties of Matrix
Operations
2.3
The Inverse of a Matrix
2.4
Elementary Matrices
2.5
Applications of Matrix
Operations
CHAPTER OBJECTIVES
■
Write a system of linear equations represented by a matrix, as well as write the matrix form
of a system of linear equations.
■
Write and solve a system of linear equations in the form
■
Use properties of matrix operations to solve matrix equations.
■
Find the transpose of a matrix, the inverse of a matrix, and the inverse of a matrix product
(if they exist).
■
Factor a matrix into a product of elementary matrices, and determine when they are
invertible.
■
Find and use the
-factorization of a matrix to solve a system of linear equations.
■
Use a stochastic matrix to measure consumer preference.
■
Use matrix multiplication to encode and decode messages.
■
Use matrix algebra to analyze economic systems (Leontief input-output models).
■
Use the method of least squares to find the least squares regression line for a set of data.
LU
A
x
±
b
.
Operations with Matrices
In Section 1.2 you used matrices to solve systems of linear equations. Matrices, however,
can be used to do much more than that. There is a rich mathematical theory of matrices,
and its applications are numerous. This section and the next introduce some fundamentals
of matrix theory.
It is standard mathematical convention to represent matrices in any one of the following
three ways.
1. A matrix can be denoted by an uppercase letter such as
2. A matrix can be denoted by a representative element enclosed in brackets, such as
±
a
ij
²
,
±
b
²
,
±
c
²
, .
.
. .
A
,
B
,
C
, .
.
. .
2.1

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*Sign up*Section 2.1
Operations with Matrices
47
3. A matrix can be denoted by a rectangular array of numbers
As mentioned in Chapter 1, the matrices in this text are primarily
real matrices.
That is,
their entries contain real numbers.
Two matrices are said to be
equal
if their corresponding entries are equal.
Consider the four matrices
and
Matrices
and
are
not
equal because they are of different sizes. Similarly,
and
are
not equal. Matrices
and
are equal if and only if
REMARK
:
The phrase “if and only if” means the statement is true in both directions. For
example, “ if and only if ” means that
implies
and
implies
A matrix that has only one column, such as matrix
in Example 1, is called a
column
matrix
or
column vector.
Similarly, a matrix that has only one row, such as matrix
in
Example 1, is called a
row matrix
or
row vector.
Boldface lowercase letters are often used
to designate column matrices and row matrices. For instance, matrix
in Example 1 can be
partitioned into the two column matrices
and
as follows.
A
±
±
1
3
2
4
²
±
±
1
3
±
±
2
4
²
±
³
a
1
±
a
2
´
a
2
±
±
2
4
²
,
a
1
±
±
1
3
²
A
C
B
p
.
q
q
p
q
p
x
±
3.
D
A
C
B
B
A
D
±
±
1
x
2
4
²
.
C
±
³
13
´
,
B
±
±
1
3
²
,
A
±
±
1
3
2
4
²
,
EXAMPLE 1
Equality of Matrices
±
a
11
a
21
a
31
.

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