WarmUp

1
Warm-Up
6 minutes
Write each system as a matrix equation. Then
solve the system, if possible, by using the
matrix equation.
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4.5.1 Using Matrix Row Operations

• Objectives
• Represent a system of equations as an augmented
  matrix
• Perform elementary row operations on matrices

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Matrix Row Operations
The row-reduction method of solving a system
allows you to determine whether the system is
independent, dependent, or inconsistent.
The row-reduction method of solving a system is
performed on an augmented matrix.
An augmented matrix consists of the coefficients
and constant terms in the system of equations.
System
Augmented Matrix
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Matrix Row Operations
The goal of the row-reduction method is to
transform, if possible, the coefficient columns
into columns that form an identity matrix.
This is called the reduced row-echelon form of an
augmented matrix if the matrix represents an
independent system.
The resulting constants will represent the unique
solution to the system.
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Elementary Row Operations
The following operations produce equivalent
matrices, and may be used in any order and as
many times as necessary to obtain reduced
row-echelon form.
-Interchange two rows
-Multiply all entries in one row by a nonzero
number
-Add a multiple of one row to another row
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Row Operations and their Notations
-Interchange rows 1 and 2
-Multiply each entry in row 3 by -2
-Replace row 1 with the sum of row 1 and 4 times
each entry in row 2
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Example 1
Perform the indicated row operations on matrix A.
11
19
20
14
-11
6
-7
5
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Practice
Perform the indicated row operations on matrix A.
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Homework
p.256 8-12
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Warm-Up
6 minutes
Perform the indicated row operations on matrix A.
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4.5.2 Using Matrix Row Operations

• Objectives
• Solve a system of linear equations by using
  elementary row operations

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Example 1
Solve the system of equations by using the
row-reduction method. Then classify the system.
1
2
16
0
-3
-21
x 2 y 7
independent
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Practice
Solve the system of equations by using the
row-reduction method. Then classify the system.
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Example 2
Solve the system of equations by using the
row-reduction method. Then classify the system.
x 1.4z 0
y 0.2z 0
0 1
no solution, inconsistent
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Example 3
Solve the system of equations by using the
row-reduction method. Then classify the system.
x z -1
y 2z 3
0 0
infinitely many solutions
dependent
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Practice
Solve the system of equations by using the
row-reduction method. Then classify the system.
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Homework
worksheet