College Algebra 9th txtbk

SECTION 7–2 Solving Systems of Linear Equations Using Gauss–Jordan Elimination 443
Coefficient
matrix
1 5 3
£ 6 0 4 §
2 3 4
Z Figure 2
Constant
matrix
4
£ 1 §
7
Now we turn our attention to the connection between matrices and systems of equations.
Consider the system of equations
x 5y 3z 4
6x 4z 1
2x 3y 4z 7
If we remove the variables and leave behind the numbers, we can think of the result as a matrix:
1 5 3 4
£ 6 0 4 † 1 §
2 3 4 7
This is known as the augmented coefficient matrix for the system. We can also define the
coefficient matrix and the constant matrix for the system, as shown in Figure 2. The augmented
coefficient matrix contains all of the information about the system needed to solve
it. Note that we put in a coefficient of zero for the missing y in the second equation, and
that we drew a vertical bar to separate the coefficients from the constants. (Matrices displayed
on a graphing calculator won’t have that line.)
Since we would like to be able to use matrices to solve large systems with many variables,
moving forward we will use x 1 , x 2 , x 3 , and the like, rather than x, y, z, and so on. In
this notation, we will rewrite system (2) as
x 1 5x 2 3x 3 4
6x 1 4x 3 1
2x 1 3x 2 4x 3 7
In Section 7-1, we used E i to denote the equations in a linear system. Now we use R i to
denote the rows and C i to denote the columns, respectively, in a matrix, as illustrated below
for system (2).
(2)
C 1 C 2 C 3 C 4
1 5 3 4 R 1
£ 6 0 4 † 1 § R 2
(3)
2 3 4 7
Our goal will be to learn how to perform the basic steps we used to solve systems using
elimination by addition, but on an augmented matrix. This enables us to focus on the numbers
without being concerned about algebraic manipulations.
R 3
EXAMPLE 1 Writing an Augmented Coefficient Matrix
Write the augmented coefficient matrix corresponding to each of the following systems.
(A) 2x 1 4x 2 5 (B) 3x 1 2x 3 4 (C) 2x 1 x 2 4
3x 1 x 2 6 7x 1 5x 2 3x 3 0 3x 1 5x 3 6
2x 2 x 3 3
SOLUTIONS
(A) c 2 4
(B) c 3 0 2 4 ` (C)
3 1 ` 5
6 d
7 5 3 0 d
2 1 0 4
£ 3 0 5 † 6 §
0 2 1 3
MATCHED PROBLEM 1
Write the augmented coefficient matrix corresponding to each of the following systems.
(A) x 1 2x 2 3 (B) 2x 2 2x 3 4 (C) 2x 1 x 2 x 3 4
3x 1 5x 2 8 7x 1 5x 2 3x 3 0 3x 1 4x 2 6
x 1 5x 3 3