Transcript Chapter 6

Chapter 6
Section 6.1
Systems of Linear Equations
Section 6.1
Systems of Linear Equations
•
•
•
•
•
Equations
Linear equations (1st degree equations)
System of linear equations
Solution of the system
System of 2 linear equations in 2 variables:
independent, dependent, inconsistent systems
• Solution methods: substitution and elimination.
• Example 1 (p. 313), 2, 4 (p.314), 6 (p. 316)
Figure 2
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Figure 3
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Figure 4
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Education, Inc.. All rights
6.2 Larger system of linear equations
Two systems are equivalent if they have the
same solutions.
Elementary operations (to produce an
equivalent system):
1. Interchange any two equations
2. Multiply both sides of an equation by a
non-zero constant.
3. Replace an equation by the sum of itself
and a constant multiple of another
equation in the system.
Elimination method
Example 1 (p. 320)
Elimination method for solving larger system of
linear equations:
1. Make the leading coefficient of the first
equation 1.
2. Eliminate the leading variable of the first
equation from each later equation.
3. Repeat steps 1 and 2 for the second equation.
4. Repeat steps 1 and 2 for the third, fourth
equation and so on, till the last equation.
5. Then solve the resulting system by back
substitution.
MATRIX METHODS
• Matrix
• Row, Column, Element (entry)
• Augmented matrix
Row operations on matrices:
1. Interchange any two rows.
2. Multiply each element of a row by a nonzero constant.
3. Replace a row by the sum of itself and a
constant multiple of another row of the
matrix.
Example 2 (p. 322)
MATRIX METHODS
Row echelon form:
1. All rows having entirely zeros (if any) are at
the bottom
2. The first nonzero entry in each row is 1
(called leading 1).
3. Each leading 1 appears to the right of the
leading 1’s in any preceding rows.
Example:
1  2 4 6


0 1 3 0 
0 0 1 2 


DEPENDENT AND
INCONSISTENT SYSTEMS
• Example 9: 2 X  3Y  4Z  6

 X  2Y  Z  9
• Solution: The system has infinitely many
solutions (the system is dependent)
• Example 11: 4 X  12Y  8Z  4
 2 X  8Y  5Z  0
 3 X  9Y  6 Z  2

• Solution: the system has no solution (it is
inconsistent)
GAUSS-JORDAN METHOD
Example 1:
 x  5 z  6  y

3 x  3 y  10  z
 x  3y  2z  5

(The system is independent)
GAUSS-JORDAN METHOD
Example 2:
2 x  4 y  4

 3x  6 y  8
 2x  y  7

(The system is inconsistent)
GAUSS-JORDAN METHOD
Example 3:
 x  2 y  3z  0

3x  2 y  z  6
(The system is dependent)
GAUSS-JORDAN METHOD
A matrix is said to be in reduced row echelon
form if it is in row echelon form and every column
containing a leading 1 has zeros in all its other
entries.
Example:
1 0 0 6


0 1 0 3 
0 0 1 2 


GAUSS-JORDAN METHOD
1. Arrange the equations with the variables terms in
the same order on the left of the equal sign and
the constants on the right.
2. Write the augmented matrix of the system.
3. Use the row operations to transform the
augmented matrix into reduced row echelon form:
4. Stop the process in step 3 if you obtain a row
whose elements are all zeros except the last one.
In that case, the system is inconsistent and has no
solutions. Otherwise, finish step 3 and read the
solutions of the system from the final matrix.
6.3 Applications of Systems of Linear
Equations
Example 1: (p. 333)
A company plans to spend $3 million on 200 new
vehicles. Each van will cost $10000, each small
truck $15000, and each large truck $25000. Past
experience shows that the company needs twice
as many vans as small trucks. How many of each
kind of vehicles can the company buy?
6.3 Applications of Systems of Linear
Equations
Example 2: (p. 334)
Ellen plans to invest a total of $100000 in a money market
account, a bond fund, an international stock fund, and a
domestic stock fund. She wants 60% of her investment to
be conservative (money market and bonds). She wants the
amount in international stocks to be one-forth of the amount
in domestic stocks. Finally, she needs an annual return of
$4000. Assuming she gets annual return of 2.5% on the
money market account, 3.5% on the bond fund, 5% on the
international stock fund, and 6% on the domestic stock
fund, how much should she put in each investment?
6.3 Applications of Systems of Linear
Equations
Example 3:
An animal feed is to be made from corn, soybean,
and cottonseed. Determine how many units of
each ingredient are needed to make a feed that
supplies 1800 units of fiber, 2800 units of fat, and
2200 units of protein, given the information below:
Corn Soybean Cottonseed Totals
Fiber
Fat
Protein
10
30
20
20
20
40
30
40
25
1800
2800
2200
6.3 Applications of Systems of Linear Equations
Example 4:
The concentrations (in parts per million) of carbon dioxide
(a greenhouse gas) have been measured at Mauna Loa,
Hawaii, since 1959. The concentrations are known to have
increased quadratically. The following table lists readings
for 3 years:
Year
1964
1984
2004
Carbon
Dioxide
319
344
377
a) Use the given data to construct a quadratic function that
gives the concentration in year x
b) Use this model to estimate the carbon dioxide
concentrations in 2010 and 2014.
6.3 Applications of Systems of Linear
Equations
Example 5:
Kelly Karpet Kleaners sells rug-cleaning machines.
The EZ model weighs 10 pounds and comes in a
10-cubic-foot box. The compact model weighs 20
pounds and comes in an 8-cubic-foot box. The
commercial model weighs 60 pounds and comes in
a 28-cubic-foot box. Each of Kelly’s delivery van
has 248 cubic feet of space and can hold a
maximum of 440 pounds. In order for a van to be
fully loaded, how many of each model should it
carry?
6.4 Basic Matrix Operations
•
•
•
•
•
Size of a matrix
Row matrix
Column matrix
Square matrix
Element of matrix A:
aij : element in row i and column j
Sum of two matrices
• Sum of two matrices of the same size:
Given matrices X and Y (both have the
same size m  n). Matrix Z = X + Y has
elements zij = xij + yij, where xij , yij, zij are
the elements on the i-th row, j-th column of
matrices X, Y and Z.
5  6  4 6   1 0
8 9    8  3  16 6

 
 

• Additive inverse of a matrix A is the
matrix –A in which each element is the
additive inverse of the corresponding
element of A.
1 2 3
A 
,

 4 5 6
  1  2  3
A 


4

5

6


• Zero matrix O: all elements are zeros.
• Identity property:
A + O = O + A = A, A is any matrix.
• Subtraction:
The difference of X and Y (same size) is
matrix Z, in which each element is the
difference of the corresponding elements
of X and Y, or, equivalently:
Z = X – Y = X + (– Y)
5  6  4 6  9  12
8 9    8  3  0 12 

 
 

• Product of a scalar k and a matrix X is the
matrix kX, each of whose elements is k
times the corresponding element of X.
 4 6  12  18
(3) 



8

3
24
9

 

Exercise:
 2 4
  6 2
• Let
A 
and B  


0
3
4
0




Find each of the following:
1. 2A
2. –3B
3. 3A – 10B
6.5 MATRIX PRODUCT AND INVERSE
Product of a Row Matrix and a Column Matrix
Matrix Product
If A is an m × p matrix and B is a p × n matrix,
then the matrix product of A and B, denoted
AB, is an m × n matrix whose element in the ith
row and jth column is the real number obtained
from the product of the ith row of A and the
jth column of B.
If the number of columns in A does not equal
the number of rows in B, then the matrix
product AB is not defined.
Check Sizes Before Multiplication
MATRIX PRODUCT
7-1-67
Example
Product (Sigma Notation)
• Let A be an mn matrix and let B be an nk
matrix. The product matrix AB (denoted C) is the
mk matrix whose entry in the i-th row and j-th
column is:
n
Cij =  Ail  Blj
l 1
1 2 3 10 1 10  2  20  3  30 140
2 1 0  20  
   40 


   
  
4 0 2 30 
 100

Properties
• Associative property:
A(BC) = (AB)C, A+(B+C) = (A+B)+C
• Distributive property:
A(B+C) = AB + AC
• Identity matrix I:
On the main diagonal: all elements are 1
Elsewhere: all elements are 0
• Not commutative: AB  BA in general
Definition of inverse matrix:
• Given matrix A, if exists matrix B so that
AB = I, B is called inverse matrix, and
denoted A-1 (read A-inverse).
• Singular, non-singular matrix
Inverse matrix calculation:
1. Form the augmented matrix [A| I]
2. Perform row operations on [A| I] to get a
matrix of the form [I | B].
3. Matrix B is A-1.
6.6 Applications of Matrices
1. Solving systems with matrices:
System AX = B, where A is coefficient
matrix, X is the matrix of variables, and B is
the matrix of constants, is solved by first
finding A-1. Then, if A-1 exists, X = A-1B.
Example:
2x – 3y = 4
x + 5y = 2
Write matrices A, X, B in this example.
6.6 Applications of Matrices
2. Input-output analysis
• Input-output matrix A (or technological
matrix) of an economy.
Example 3.
Agriculture Manufacturing Transporta tion
Agricultur
e 
Manufactur
ing 
Transporta
tion
0
1/ 2
1/ 4
1/ 4
0
1/ 4
1/ 3
1/ 4
0

A


6.6 Applications of Matrices
2. Input-output analysis
• Production matrix X
• Demand matrix D = X – AX
Example 4.
 0 1 / 4 1 / 3 60 29
60
X  52
48






AX  1 / 2 0 1 / 4  52  42
1 / 4 1 / 4 0  48 28
60 29  31
D  X  AX  52  42  10
48  28 20
6.6 Applications of Matrices
2. Input-output analysis
• In practice, A and D are known, we need to
find the production matrix: X–1 = (I – A) –1D
Example 6: An economy depends on 2 basic
products: wheat and oil. To produce 1 ton of
wheat requires .25 ton of wheat and .33 ton of
oil. The production of 1 ton of oil consumes
.08 ton of wheat and .11 ton of oil. Find the
production that will satisfy the demand of 500
ton of wheat and 1000 ton of oil.