Transcript Row-Echelon Reduction - Dr. Taylor Math Coach

3/27/2016
Agenda
• Textbook / Web Based Resource
– Basics of Matrices
– Row-Echelon Form
– Reduced Row Echelon Form
• Classwork
INSERT HERE
• Homework
INSERT HERE
3/27/2016
Matrices &
Linear Systems
3/27/2016
By the End of the Day:
You should know how to:
•
•
•
•
•
Determine the Order of a matrix
Perform Elementary Row Operations
Identify Elementary Row Operation
Identify if a Matrix is in Row-Echelon Form
Identify if a Matrix is in Reduced RowEchelon Form
3/27/2016
Definitions
Matrix:
a rectangular array of numbers. Each number is
called an entry
Order of a matrix:
Tells you how many rows and columns the
matrix has. If a matrix has m rows and n
columns then it is an m x n matrix.
Square matrix
Matrix which has the same number of rows and
columns
3/27/2016
Order of a Matrix
Do (FROM TEXT)
 2

3 rows  4

  8
2
2
 1 3
5 8
 6  7 1
Dimension: 1 x 2
Dimension: 5 x 1
This is a 3 x 4 matrix
Dimension: 3 x 3
The 2,3 entry is:
5
Dimension: 1 x 1
3/27/2016
Row Operations
There are 3 Elementary Row Operations:
1. Interchange two rows
2. Multiply a row by a non zero constant
3. Add a multiple of a row to another row
3/27/2016
Row Operations
1. Interchange 2 Rows:
Interchange R1 and R3:
 2  2  1 3
4

2
5
8


 8  6  7 1
 8  6  7 1
4

2
5
8


 2  2  1 3
3/27/2016
Row Operations
2. Multiply a row by a non zero constant:
Multiply R2 by -2:
 2  2  1 3
4

2
5
8


 8  6  7 1
3 
 2  2 1
 8  4  10  16


 8  6  7
1 
3/27/2016
Row Operations
3. Add a multiple of 1 row to another row:
Add 3 * R1 to R2
 2  2  1 3
4

2
5
8


 8  6  7 1
 2  2 1 3 
 10  4 2 17


 8  6  7 1 
3(2) + 4 = 10
3(-1) + 5 = 2
3(-2) + 2 = -4
3(3) + 8 = 17
3/27/2016
Row Operations
To Determine what row operations occurred:
• Identify which row changed
– If 2 changed, they were probably interchanged
• Determine if each number is the result of
multiplying each number by a constant
– Determine the constant
• Determine how much was added to each entry
to get the row.
– Determine which row must have been multiplied to
get these numbers, and by how much.
3/27/2016
Row Operations
What row operation was performed?
 2  2  1 3
 2  2  1 3
4

4

2
5
8
2
5
8




 4  10  9 7
 8  6  7 1
Which row changed? R3
Was it mult by a constant? No
How much was added? 4, -4, -2, 6
Which row was multiplied? R1 (by 2)
What is the row operation? Add 2 * R1 to R3
3/27/2016
Row Operations
(Examples of Row-Echelon operation notation…
…may be necessary for more than one operation)
3*R1 + R2
or
2*R1 + R2
5*R1 + R3
etc…
3/27/2016
Row-Echelon Form
A matrix is in row echelon form if:
1. Any rows consisting of entirely zeros occur
at the bottom of the matrix
2. For each row that does not consist entirely
of zeros, the first nonzero entry is a 1
3. As you work down the matrix the “leading 1”
moves to the right.
Still Makes the “Staircase”
3/27/2016
Row-Echelon Form
Row Echelon Form
Not Row-Echelon Form
2 2
5
1 54  5
0 4 3

2
1
0
1
4

2
3


0 0
1 104 146
3/27/2016
Row-Echelon Form
• A matrix is in reduced row echelon form if
every entry above and below a leading 1 is 0.
Reduced Row Echelon
Form
1 0 0 5 
0 1 0 4 


0 0 1  2
Not Reduced RowEchelon Form
1 4  3 5 
0 1  5 4 


0 0 1  2
3/27/2016
Row-Echelon Form
You Try These on Your Own:
1. INSERT YOUR PROBLEMS HERE
5.1a
3/27/2016
It’s the End of the Day:



Do you know how to:
• Determine the Order of a matrix?
• Perform Elementary Row Operations?
• Identify Elementary Row Operation?
• Identify if a Matrix is in Row-Echelon
Form?
• Identify if a Matrix is in Reduced RowEchelon Form?


3/27/2016
Homework
Study:
INSERT HERE
Do:
INSERT HERE
Read & Take Notes:
INSERT HERE
3/27/2016
Resource Credits
Justin Bohannon