10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
10.2 Systems of Linear Equations: Matrices Day 1
Objectives:
1. Write the augmented matrix of a system of linear equations. 2. Write the system from the augmented matrix.
3. Perform row operations on a matrix.
4. Solve a system of linear equations using matrices. Nov 9­12:20 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
What is......a matrix?
Definition:
A matrix (plural matrices) is a rectangular
array of numbers written within brackets.
Dec 2­2:44 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Augmented Matrix A matrix used to represent a system of linear equations.
System
{
x + 4y = 14
3x ­ 2y = 0
Augmented Matrix
[ ]
1 4 14
3 ­2 0
Dec 2­3:03 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Write the Augmented Matrix of a System of Linear Equations
Example #2:
Example #1:
{
3x ­ 4y = ­6
2x ­ 3y = ­5
{
2x ­ y + z = 0
x + z ­ 1 = 0
x + 2y ­ 8 = 0
Nov 30­12:37 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Write the System from the Augmented Matrix Example #4:
Example #3:
[ ]
5 2 13
­3 1 ­10
[ ]
3 ­1 ­1 7
2 0 2 8
0 1 1 0
Nov 30­12:38 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Perform Row Operations on a Matrix Row Operations: Manipulations on an augmented matrix that are used to solve systems of equations.
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Perform Row Operations on a Matrix Three basic row operations:
1. Interchange any two rows.
2. Replace a row by a nonzero multiple of that row.
3. Replace a row by the sum of that row and a constant nonzero multiple of some other row.
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Before we get too much further...
what are rows and columns?
[ ]
1 2 3
4 5 6
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Perform Row Operations on a Matrix Example #5:
Apply the row operation R2 = ­4r1 + r2 to the following matrix.
[ ]
1 2 3
4 ­1 2
Answer:
[ ]
1 2 3
0 ­9 ­10
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Perform Row Operations on a Matrix Example #6:
Apply the row operation R2 = ­3r1 + r2 to the following matrix.
[ ]
1 ­2 2
3 ­5 9
Answer:
[ ]
1 ­2 2
0 1 3
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Perform Row Operations on a Matrix Example #7:
Perform each row operation on the given augmented matrix.
[ ]
1 ­3 2 ­6
2 ­5 3 ­4
­3 ­6 4 6
(a) R2 = ­2r1 + r2
(b) R3 = 3r1 + r3
[ ]
(b) [ ]
(a)
1 ­3 2 ­6
0 1 ­1 8
­3 ­6 4 6
1 ­3 2 ­6
2 ­5 3 ­4
0 ­15 10 ­12
Nov 30­12:39 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Solve a System of Linear Equations Using Matrices How do we solve a system of linear equations?
Use row operations to get to...
row
echelon
form
Example:
[ ]
1 ­1 1 8
0 1 ­12 ­15
0 0 1 1
Reduced row echelon
[ Nov 30­12:40 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Solve a System of Linear Equations Using Matrices Example #8: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
[ ]
1 0 0 4
0 1 0 ­3
0 0 1 1
Answer:
Nov 30­12:40 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Solve a System of Linear Equations Using Matrices Example #9: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
[ ]
1 0 0 ­5
0 1 0 4
0 0 0 3
Answer:
Nov 30­12:40 PM
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10.2 day 1 Systems of Linear Equations ­ Matrices 2010 December 03, 2010
Homework:
page 754 (6 ­ 24 even, 25 ­ 28) Nov 30­12:40 PM
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