Chapter 2 – Part 2
MATRICES
A: Augmented Matrices and Row Operations
(Lessons 2.2 pg 68 - 70)
Augmented Matrices
Suppose you are given a system of The system can be written as a matrix:
equations such as:
 
To separate the coefficients of the variables from the constants after the equals signs, we draw in a vertical line in the matrix. This is called an augmented matrix.
Example 1: Write an augmented matrix for each system of equations. Do not solve.
a)  b) 
c) 
Example 2: Write the system of equations associated with each augmented matrix.
a)  b) 
(example 2 continued…. Write the system of equations associated with each augmented matrix)
c)  d) 
In algebra, our goal when faced with a system of equations is to find a solution for x, y, and z that makes the system of equations true.
Note that when you see this pattern:  You end up with 
(1’s on main diagonal, 0’s elsewhere, constants last column)
Row Operations
We will be using “Row Operations” to manipulate matrices to help us solve systems of equations.
What you are allowed to do:
1. INTERCHANGE TWO ROWS
2. MULTIPLY THE ELEMENTS OF A ROW BY A NONZERO REAL NUMBER
3. ADD A NONZERO MULTIPLE OF THE ELEMENTS OF ONE ROW TO THE CORRESPONDING ELEMENTS OF A
NONZERO MULTIPLE OF SOME OTHER ROW.
Example 3: Use the indicated row operations to change each matrix.
a) Interchange R1 with R2.
(Example 3 continued: Use the indicated row operation to change each matrix)
b) Replace R3 by R3 .
c) Replace R2 with (-2)R1 +R2
d) Replace R3 with (-3)R2 + 5R3
e) Replace R1 with (-2)R3 + 3R1
f) Replace R2 with (-7)R3 + 6R2
g) Replace R2 with (-1)R1 +4R2
B: Gauss-Jordan Method for Solving Systems of Equations
(Lesson 2.2 , textbook pg 70 – 80)
Problem: Find the solution to a system of equations like 
Strategy:
1. Write the system of equations as an augmented matrix
2. Use Row Operations to transform the matrix into a matrix with whole numbers on the main diagonal, but 0’s
elsewhere.
3. Use Row Operations to transform the matrix into an “identity” matrix.
(1’s on diagonal, 0’s elsewhere)
4. Final solution = numbers in the “answer” column of the matrix.
Example:  Example: 
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