chapter 6 section 6.1 systems of linear equations
TRANSCRIPT
Chapter 6
Section 6.1
Systems of Linear Equations
Section 6.1Systems 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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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 matrixRow operations on matrices:1. Interchange any two rows.2. Multiply each element of a row by a non-
zero 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 METHODSRow echelon form:1. All rows having entirely zeros (if any) are at
the bottom2. 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:
2
0
6
100
310
421
DEPENDENT AND INCONSISTENT SYSTEMS
• Example 9:
• Solution: The system has infinitely many solutions (the system is dependent)
• Example 11:
• Solution: the system has no solution (it is inconsistent)
92
6432
ZYX
ZYX
2693
0582
48124
ZYX
ZYX
ZYX
GAUSS-JORDAN METHOD
Example 1:
523
1033
65
zyx
zyx
yzx
(The system is independent)
GAUSS-JORDAN METHOD
Example 2:
72
863
442
yx
yx
yx
(The system is inconsistent)
GAUSS-JORDAN METHOD
Example 3:
623
032
zyx
zyx
(The system is dependent)
GAUSS-JORDAN METHODA 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:
2
3
6
100
010
001
GAUSS-JORDAN METHOD1. 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.
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:
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
Year 1964 1984 2004
Carbon Dioxide
319 344 377
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?
6.3 Applications of Systems of Linear Equations
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.
616
01
38
64
98
65
• Additive inverse of a matrix A is the matrix –A in which each element is the additive inverse of the corresponding element of A.
• Zero matrix O: all elements are zeros.
• Identity property:
A + O = O + A = A, A is any matrix.
654
321,
654
321AA
• 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)
120
129
38
64
98
65
• 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.
Exercise:• Let
Find each of the following:1. 2A 2. –3B 3. 3A – 10B
924
1812
38
64)3(
04
26
30
42BandA
Product of a Row Matrix and a Column Matrix
6.5 MATRIX PRODUCT AND INVERSE
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 mn matrix and let B be an nk
matrix. The product matrix AB (denoted C) is the mk matrix whose entry in the i-th row and j-th column is:
Cij =
n
lljil BA
1
100
40
140303202101
30
20
10
204
012
321
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 1Elsewhere: 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 matrixInverse 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.
A
tionTransporta
ingManufactur
eAgriculturtionTransportaingManufactureAgricultur
04/14/1
4/102/1
3/14/10
6.6 Applications of Matrices 2. Input-output analysis
• Production matrix X
• Demand matrix D = X – AX
Example 4.
48
52
60
X
28
42
29
48
52
60
04/14/1
4/102/1
3/14/10
AX
20
10
31
28
42
29
48
52
60
AXXD
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.