Systems of Linear Equations & Matrices
Sections 4.2 & 4.3
After today’s lesson, you will be able to
 Use terms associated with matrices.
 Set up and solve the augmented matrix associated with a
linear system in two variables on a calculator.
 Identify the three possible matrix solution types for a linear
system in two variables.
 Convert a matrix to reduced row echelon form using the
calculator.
1
Matrix Methods
It is impractical to solve more complicated linear
systems by hand. Computers and calculators now
have built in routines to solve larger and more
complex systems.
Matrices, in conjunction with graphing utilities
and or computers are used for solving more
complex systems.
2
Matrices
A matrix is a rectangular array of numbers written within
brackets.
The subscripts give the “address” of each entry of the matrix.
For example the entry a23 is found in the second row and third
column
 a11 a12
a
 21 a22
a13
a23
a14 

a24 
Since this matrix has 2 rows and 4 columns, the dimensions of
the matrix are 2 x 4.
Each number in the matrix is called an element.
3
Matrix Solution of Linear Systems
We can represent a linear
system of equations by an
augmented matrix, a
matrix which stores the
coefficients and constants
of the linear system and
then manipulate the
augmented matrix to obtain
the solution of the system.
Make sure the system’s
equations are in general form
before creating a matrix.
Example: Write the augmented
matrix for the system:
3x  5 y  9

6 x  y  2
The augmented matrix associated
with the above system is:
 3 5 9
 6 1 2


4
Reduced Row Echelon Form
To solve the system of linear equations, we want to convert
our augmented matrix to a matrix in reduced row-echelon
form. We will use our graphing calculators to do this.
An RREF matrix has 1’s down its main diagonal and 0’s
below and above the 1’s.
The reduced row-echelon form of the previous matrix is:
1

1 0  33 


 0 1 20 

11 
See handout for directions on
how to obtain the rref matrix
using your calculator.
5
Writing a Solution from a Matrix in
Reduced Row Echelon Form
Example: Write the solution to the system represented by the
each of the following matrices.
1)
1 0 3
0 1 17 


1 0 0 13.2 


2) 0 1 0 15.6 
0 0 1 8.7 
6
Using Subscripted Variables
Sometimes, instead of using different letters to represent the
unknown quantities, we use the same variable with different
subscripts.
Example: Solve the system using matrices. Round to hundredths.
5.70 x1  8.55 x2  35.91

4.50 x1  5.73x2  76.17
a) Write the augmented matrix for the system.
b) Using your calculator, find the RREF matrix for the system.
c) Write your conclusion about the system.
7
Continued
5.70 x1  8.55 x2  35.91

4.50 x1  5.73x2  76.17
8
Example
Example: Given the following system, complete the following
parts.
x  2 y  2  z

2 x  2 y  5  z
2 z  y  x  7

a) Write the augmented matrix for the system
9
Example
b) Using your calculator, find the RREF matrix for the system
c) Write your conclusion about the system.
10
Example
Example: Given the following system, complete the following
parts.
10 x  2 y  6

 y  5x  3
a) Write the augmented matrix for the system
b) Using your calculator, find the RREF matrix for the system
c) Write your conclusion about the system.
11
Continued
12
Example
Example: Given the following system, complete the following
parts.
 x  y  3

 2 y  z  1
5 x  z  1

a) Write the augmented matrix for the system
b) Using your calculator, find the RREF matrix for the system
c) Write your conclusion about the system.
13
Continued
14
Possible Final Matrix Forms for a
Linear System in Two Variables
Form 1: Unique Solution
(Consistent and Independent)
1 0 a 
0 1 b 


Form 2: Infinitely Many Solutions
(Consistent and Dependent)
1 a b 
0 0 0 


Form 3: No Solution
(Inconsistent and Independent)
1 a b 
0 0 c 


15