Book Title: eTextbook: Finite Mathematics and Applied Calculus
4.2. Using Matrices to Solve Systems of Equations
4.2. Using Matrices to Solve Systems of
Equations
The Augmented Matrix of a System of Linear
Equations
In this section we describe a systematic method for solving systems of
equations that makes solving large systems of equations in any number of
unknowns straightforward. Although this method may seem a little
cumbersome at first, it will prove immensely useful in this and the next several
chapters. First, we introduce some terminology.
Linear Equation
A linear equation in the n variables x
a1x1 + ⋯ + anxn = b
The numbers a
, a2, … , an
1
1,
x2, … , xn
has the form
(a1, a2, … , an, b constants)
are called the coefficients, and the number b is
called the constant term, or right-hand side.
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Quick Examples
1.
3x − 5y = 0
2.
x + 2y − z = 6
4.
30x1 + 18x2 + x3 + x4 = 19
Linear equation in x and y Coef f icients: 3, −5; co
Linear equation in x, y, z Coef f icients: 1, 2, −1;
Linear equation in x1, x2, x3, x4 Coe
NOTE When the number of variables is small, we will almost always use
x, y, z, …
(as in Quick Examples 1 and 2) rather than x
1,
x2, x3, …
as the
names of the variables.
Notice that a linear equation in any number of unknowns (for example,
2x − y = 3
) is entirely determined by its coefficients and its constant term. In
other words, if we were simply given the row of numbers
[2
− 1
3],
we could easily reconstruct the original linear equation by multiplying the first
number by x , multiplying the second by y , and inserting a plus sign and an
equals sign, as follows:
2 ⋅ x + (−1) ⋅ y = 3
or
2x − y = 3.
Similarly, the equation
−4x + 2y = 0
is represented by the row
[−4
2
0],
and the equation
1
−3y =
4
is represented by the row
1
[0
− 3
].
4
As the last example shows, the first number is always the coefficient of x , and
the second is the coefficient of y . If an x or a y is missing, we write a zero for its
coefficient. We shall call such a row the coefficient row of an equation.
If we have a system of equations, for example, the system
2x −
y = 3
−x + 2y = −4,
we can put the coefficient rows together like this:
2 −1
3
[
].
−1
276
2 −4
We call this the augmented matrix of the system of equations. The term
“augmented” means that we have included the right-hand sides 3 and −4 . We
will often drop the word “augmented” and simply refer to the matrix of the
system. A matrix (plural: matrices) is nothing more than a rectangular array
of numbers, as above.
Matrix, Augmented Matrix
A matrix is a rectangular array of numbers. The augmented matrix of a
system of linear equations is the matrix whose rows are the coefficient
rows of the equations.
Quick Example
4.
The augmented matrix of the system
x + y = 3
x − y = 1
1
1
3
is [
] .
1
−1
1
We’ll be studying matrices in more detail in Chapter 4.
Q:
What good are coefficient rows and matrices?
A:
Think about what we do when we multiply both sides of an equation by a
number. For example, consider multiplying both sides of the equation
2x − y = 3
by −2 to get −4x + 2y = −6 . All we are really doing is multiplying the
coefficients and the right-hand side by −2 . This corresponds to multiplying the
row [2 − 13] by −2 , that is, multiplying every number in the row by −2 . We
shall see that any manipulation we want to do with equations can be done
instead with rows. This fact leads to a method of solving equations that is
systematic and generalizes easily to larger systems.
Here is the same operation in both the language of equations and the language
of rows. (We refer to the equation here as Equation 1, or simply E for short,
1
and to the row as Row 1, or R .)
1
Equation
Multiply by −2
E1:
2x −
Row
y = 3
[2
−1
3]
R1
:
(−2)E1:
−4x + 2y = −6
[−4
2
−6]
(−2)R1
Multiplying both sides of an equation by the number a corresponds to multiplying
the coefficient row by a .
Now look at what we do when we add two equations.
Equation
Add:
E1:
E2:
E1 + E2:
2x −
y = 3
−x + 2y = −4
x +
y = −1
Row
[
2
− 1
3] R1
[−1
2
− 4]
[
1
− 1] R1 + R2
1
R2
All we are really doing is adding the corresponding entries in the rows, or
adding the rows. In other words,
Adding two equations corresponds to adding their coefficient rows.
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In short, the manipulations of equations that we saw in Section 4.1 can be done
more easily with rows in a matrix because we don’t have to carry x , y , and
other unnecessary notation along with us; x and y can always be inserted at the
end if desired.
The manipulations we are talking about are known as row operations. In
particular, we use three elementary row operations.
Elementary Row Operations
Type 1: Replacing R by aR (where a ≠ 0 )
i
i
In words: multiplying or dividing a row by a nonzero number.
Type 2: Replacing R by aR
i
i
± bRj
(where a ≠ 0 )
In words: multiplying a row by a nonzero number and adding or
subtracting a multiple of another row.
Type 3: Switching the order of the rows
This corresponds to switching the order in which we write the equations;
occasionally, this will be convenient.
For Types 1 and 2 we write the instruction for the row operation next to
the row we wish to replace. (See the Quick Examples below.)
⎢⎥
Quick Examples
5.
Type 1:
1 3 −4
[
]
0 4
6.
1
2
[
]
0 12
Replace R2 by 3R2.
6
Type 2:
1 3 −4
[
]
0 4
7.
→
3R2
3 −4
4 0 −22
4R1 − 3R2
→
2
[
0 4
]
Replace R1 by 4R1 −
2
Type 3:
⎡
⎣
1 3 −4
0 4
2
1 2
3
⎤
⎦
⎡
R1 ↔ R2 →
⎣
0 4
2
⎤
1 3 −4
⎦
1 2
3
Switch R1 and R2.
Using Technology
See the Technology Guide TI-83/84 Plus and Technology Guide Spreadsheet
to see how to use a TI-83/84 Plus or a spreadsheet to solve this system of
equations.
Website
www.WanerMath.com
→ Online Utilities
→ Pivot and Gauss-Jordan Tool
Enter the augmented matrix in columns x1,
x2, x3, …
. To do a row
operation, type the instruction(s) next to the row(s) you are changing as
shown, and press “Do Row Ops” once.
Details
One very important fact about the elementary row operations is that they do
not change the solutions of the corresponding system of equations. In other
words, the new system of equations that we get by applying any one of these
operations will have exactly the same solutions as the original system: It is easy
to see that numbers that make the original equations true will also make the
new equations true, because each of the elementary row operations
corresponds to a valid operation on the original equations. That any solution of
the new system is a solution of the old system follows from the fact that these
row operations are invertible: The effects of a row operation can be reversed by
applying another row operation, called its inverse. Here are some examples of
this invertibility. (Try them out in Quick Examples 5, 6, and 7.)
Operation
Inverse Operation
Replace R by 3R
2
2
Replace R by
1
R2
2
Replace R by 4R
1
1
− 3R2
Switch and R .and R .
1
2
.
3
.
Replace R by
1
3
R1 +
1
4
R2
.
4
Switch R and R .
1
2
Our objective, then, is to use row operations to change the system we are given
into one with exactly the same set of solutions in which it is easy to see what
the solutions are.