3.9: Augmented Matrices Objectives: Assignment: P. 182-3:

3.9: Augmented Matrices
Objectives:
1. To solve a linear
system by writing it in
triangular form
2. To perform elementary
row operations on an
augmented matrix
3. To solve a linear
system using an
augmented matrix
Assignment:
• P. 182-3:
– 15, 16: Triangular (show
steps)
– 17, 19: Augmented
matrix (show steps)
– 25-33, 35, 36, 38-41
• Challenge Problems
• Print Review 2b and
Checklist; Cryptography
• Matrix Project
You will be able to solve a linear
system by putting it in
triangular form
Objective 1
Exercise 1
Solve the system of equations.
x  2 y  2z  9
y  2z  5
z  3
Equivalent Systems
Two systems are equivalent if they have
the same solution.
x
 2y
x
 3y
2x
 5y
 2z

9
 4
 3z
Linear System

16
x  2 y  2z  9
y  2z  5
z  3
Equivalent Linear
System
Triangular Form
A linear system of equations is in
triangular form if:
The equations have
a stair-step pattern
The leading
coefficient of each
equation is 1
x  2 y  2z  9
y  2z  5
z  3
Triangular Form
Solving a system of equations
in triangular form is almost
too easy. The question is,
how do we rewrite a linear
system as an equivalent
system in triangular form?
The answer involves
elementary row
operations, a.k.a.
Gaussian elimination.
Money Man: Carl Gauss
(~1800s)
Elementary Row Operations
𝐴↔𝐵
1
2𝐶
2𝐴 + 𝐶
𝐴↔𝐵
1
2𝐶
2𝐴 + 𝐶
Used to create equivalent
linear systems
Interchange two
equations
Multiply one
equation by a
non-zero constant
Add a multiple of
one equation to
another equation
to replace the
latter equation
Exercise 2
Solve the system by rewriting it in triangular form.
A : x  2 y  2z 
B : x
 3y
C : 2x
 5y
9
 4
 3z 
16
2 xx 42yy 42zz
y 2 z
A+B:
2 x 5 y 3z
1. Eliminate 𝑥 in all but the first equation.
−2 A + C :
 918
 5
 16
x 2 y 2 z 
9
y 2 z 
5
y
 z  2
x 2 y 2 z  9
2. Eliminate 𝑦 in the last equation.
y 2 z  5
B+C:
z  3
Triangular Form
To rewrite a linear system in three variables
in triangular form, follow these helpful
hints:
Eliminate 𝑥 in
all but the first
equation
Step 1
Eliminate 𝑦
in the last
equation
Step 2
Use
multiplication to
make the
leading
coefficients
equal
Step 13
Exercise 3
Solve the system by rewriting it in triangular form.
2 x  4 y  2 z  16
 2 x  5 y  2 z  34
x  2 y  2z  4
You will be able to perform elementary
row operations on an augmented matrix
Objective 2
Augmented Matrix
An augmented matrix adds the column of
constant terms to the coefficient matrix.
Linear
System
x  2 y  2z
x
 3y
2x
 5y

9
 4
z

10
Coefficient
Matrix
Augmented
Matrix
 1  2 2
 1

3
0


 2  5 1
9
 1 2 2
 1

3
0

4


 2  5 1 10
Augmented Matrix
With an augmented matrix for a linear
system of equations, we can use all the
previous elementary row operations to
create an equivalent system in triangular
form. It’s just that we won’t have to worry
about those pesky variables.
Instead of using A, B, and C for our
equations, we’ll use R1, R2, and R3 for
each row of the augmented matrix.
Exercise 4
Write the system below as an augmented
matrix. Then perform elementary row
operations to rewrite it in triangular form.
x
 2y
x
 3y
2x
 5y
 2z

9
 4
z

10
R1 :  1  2 2
9
R2 :  1
3 0  4
R3 :  2  5 1 10
Exercise 4
2 4 4 18
 1 2 2 9 
R1 + R2:  0 1 2 5
 2 5 1 10 
R1 :  1  2 2
9
R2 :  1
3 0  4
R3 :  2  5 1 10
 1 2 2 9 
0

1
2
5


−2R1 + R3: 0 1 3 8
 1 2 2 9 
0

1
2
5


−R3: 0 0 1 3
 1 2 2 9 
0

1
2
5


R2 + R3: 0 0 1 3
Calculator Savvy
On a calculator, this operation can be performed
using “ref(” on a matrix.
It can be found in the
MATRIX>MATH menu
ref = reduced echelon form
Just be aware that if you
exchange rows in a system,
your calculator may give you a
different, equivalent system.
Reduced Echelon Form
Putting an augmented matrix in reduced
echelon form is elementary. Just follow
the row operations to make
look like
 1   
0 1   

.
0 0 1  
a
e

 i
b c d
f g h
j k l 
Reduced Echelon Form
Putting an augmented matrix in reduced
echelon form is elementary.
1
E
 1
 E2
  
1   
   
Eliminate the “𝑥”
element in all but
theStep
first 1row
1
E
 1
 E2

1
E3
 
  
1 z 
Eliminate the “𝑦”
element in the last
Step
row 2
Row Reduced Echelon Form
Now use row operations to further reduce
your augmented matrix.
1
E
 1
 E2

E5
1
E4
E3
1

y 
z 
Eliminate the “𝑧”
element in all but
theStep
last 3row
1
E
 1
 E2
E6
E5
1
E4
E3
1
x
y 
z 
Eliminate the “𝑦”
element in the first
Step
row 4
Exercise 5
Perform row operations to further reduce the
augmented matrix.
 1 2 2 9 
−2R3 + R2: 0 1 0 1
0 0 1 3
−2R3 + R1:  1 2 0 3
2 0 2 
0
1
0 1

0 0 1 3
2R2 + R1:  1 0 0 1
0 1 0 1


0 0 1 3
Solution
 1 2 2 9 
0

1
2
5


0 0 1 3
2 6
Row Reduced Echelon Form
Notice that when you completely row reduce
an augmented matrix, the identity matrix
replaces the coefficient matrix. What you
are left with is the solution to the system of
equations!
You will be able to solve a linear system
using an augmented matrix
Objective 3
Row Reduced Echelon Form
1
E
 1
 E2
E6
1
E5
E4
E3
1
x
y 
z 
1 0 0
0 1 0

0 0 1
Identity
Matrix
x
y 
z 
Solution
To put an augmented matrix in row reduced
echelon form, perform your Eliminations in the
following order:
Exercise 6
Solve the system using an augmented
matrix.
2 x  4 y  5z  5
x  3 y  3z  2
2x  4 y  4z  2
Why Are We Even Doing This?
This method does not
seem very efficient by
hand, especially since
there are many
opportunities for
simple arithmetic
mistakes.
Why Are We Even Doing This?
However a computer or a
calculator can
unerringly perform
these operations
quicker than you can
say “augmented
matrix.” That is why this
is the preferred method to
use to solve large
systems.
Calculator Savvy, v2.0
On a calculator, this operation can be performed
using “rref(” on a matrix.
It can be found in the
MATRIX>MATH menu
rref = row reduced echelon
form
Exercise 7
Solve the system of equations using an
augmented matrix.
4 x  y  3 z  19
 2 x  2 y  z  12
x  y  z  39
Exercise 8
Solve the system of equations using an
augmented matrix.
a  2b  6c  d  12
 2a  3b  9c  d  19
a  2b  5c  2d  15
2a  4b  12c  3d  24
The No Solution Case
Use your calculator to solve the system
below using an augmented matrix.
x  3 y  z  10
2x  6 y  2z  5
 3x  9 y  3z  7
 1
3  1 10 
 1 3  1 10


0 1
rref   2
6  2 5   0 0
  3  9

0 0

0 0
3
7




Contradiction
The Infinitely Many Case
Use your calculator to solve the system
below using an augmented matrix.
x  5 y  2 z  1
 x  2y  z  6
 2 x  7 y  3z  7
 1
5  2  1 
 1 0  1 3  28 3


rref    1  2
1 6   0 1  1 3
5 3
  2  7
0 0
3 7 
0
0

Identity
3.9: Augmented Matrices
Objectives:
1. To solve a linear
system by writing it in
triangular form
2. To perform elementary
row operations on an
augmented matrix
3. To solve a linear
system using an
augmented matrix
Assignment
• P. 182-3:
– 15, 16: Triangular
(show steps)
– 17, 19: Augmented
matrix (show steps)
– 25-33, 35, 36, 38-41
• Challenge Problems
• Print Review 2b and
Checklist; Cryptography
• Matrix Project
“Is reality augmented?”