Unit 14
Matrices and Systems of Linear Equations
In this section we will learn a method known as rowreduction for solving
SLE’s. This method works in basically the same way as the method of elim
ination, but utilizes a structure called a matrix to eliminate the repetitive
writing down of symbols which never change from one step to the next. First,
we must define what this mathematical structure called a matrix is.
Definition 14.1.
If
m
and
n
are positive integers then an
m
×
n
matrix
(
note:
m
×
n
is read “
m
by
n
”) is a rectangular array of numbers:
a
11
a
12
a
13
· · ·
a
1
n
a
21
a
22
a
23
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
a
m
3
· · ·
a
mn
in which each number
a
ij
is called the (
i, j
)
component
or (
i, j
)
entry
of the
matrix. The descriptor
m
×
n
is called the
dimension
of the matrix, where
m
is the number of
rows
(horizontal) and
n
is the number of
columns
(vertical)
in the matrix.
Notation:
Given a set of numbers
a
ij
for
i
= 1
, ..., m
and
j
= 1
, ..., n
, we
will sometimes use
A
= (
a
ij
) to denote that
A
is the
m
×
n
matrix whose
(
i, j
)entry is
a
ij
. Also, when we have a matrix called
A
, we use
a
ij
to denote
the (
i, j
)entry of matrix
A
.
That is, we use upper case letters to name
matrices, and we use the lowercase version of the same letter, subscripted
by row number (first subscript) and column number, to refer to an entry in
the matrix.
1
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Example
1
.
For
A
= (
a
ij
) =
1
2
3
4
5
6
, determine the dimension of
A
and
find
a
12
and
a
21
.
Solution:
Since
A
has 3 rows and 2 columns,
A
is a 3
×
2 matrix. Looking
in row 1, column 2, we find that
a
12
= 2. Similarly, looking in row 2 and
column 1, we see that
a
21
= 3.
A common use of matrices that we will be studying here is to represent
a system of linear equations.
Definition 14.2.
Consider any system of
m
linear equations in
n
variables,
written in standard form. Let
a
ij
be the coefficient, in the
i
th
equation, of
the
j
th
variable. The
m
×
n
matrix
A
= (
a
ij
) is called the
coefficient matrix
of the SLE.
The
augmented matrix
for a SLE is obtained by appending the
m
right hand
side values of the equations to the coefficient matrix, as an extra column in
the matrix. We always delineate this extra column in an augmented matrix
by placing a vertical line before it.
Example
2
.
Write the coefficient matrix and the augmented matrix for the
following SLE:
x

4
y
+
3
z
=
5

x
+
3
y

z
=

3
2
x

4
z
=
6
SLE in standard form
Solution:
Since the system is already in standard form, we can obtain the
coefficient matrix by simply writing the coefficients from the system in the
same order in which they appear in the SLE. For instance, we get the first
row of the coefficient matrix by extracting the coefficients of
x
,
y
and
z
, i.e.
1,

4 and 3, from the first equation.
Remember:
Our definition of the coefficient matrix and the augmented matrix
require that the system be in standard form.
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 Fall '09
 OLDS,VICKY
 Calculus, Linear Algebra, Linear Equations, Equations, Matrices, Systems Of Linear Equations, Row, SLE, Elementary matrix

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