1. nhaa/imk/sem120162017/eqt101/rk12/32
CHAPTER 2: MATRICES
Systems of Linear Equations
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2. TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
A SYSTEM WITH NO SOLUTION
A SYSTEM WITH UNIQUE SOLUTION
A SYSTEM WITH INFINITELY MANY SOLUTION
TYPES OF SOLUTIONS
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3. A SYSTEMS WITH UNIQUE SOLUTION
Consider the systems: Augmented matrix: Thus, the system has a unique solution, where
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4. A SYSTEMS WITH INFINITELY MANY SOLUTIONS
Consider the systems: Augmented matrix: Thus, the system has infinitely many solutions, by letting where s is a free variable and Then the solution for is given by
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5. A SYSTEMS WITH NO SOLUTION
Consider the systems: Augmented matrix: Thus, the system has no solution, since the coefficient of is 0.
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6. METHODS OF SOLVING SYSTEMS OF EQUATIONS
There are 4 methods can be used to solve systems of linear equations:
The Inverse of the Coefficient Matrix
Gauss Elimination
Gauss-Jordan Elimination
Cramer’s Rule
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7. METHODS OF SOLVING SYSTEMS OF EQUATIONS
The Inverse of the Coefficient Matrix
Consider a system of equations written in the form AX=B, where
The solution of X is given by:
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8. The Inverse of the Coefficient Matrix
Example Solve the system by using the inverse of the coefficient matrix: Answer:
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9. The Inverse of the Coefficient Matrix
Solution:
Write in the form AX=B
Find the inverse of A, where
Find the solution of X:
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10. The Inverse of the Coefficient Matrix
Example Solve the system by using the inverse of the coefficient matrix: Answer:
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11. The Inverse of the Coefficient Matrix
Example A four-ounce serving of Campbell’s® Chicken & Beans contains 5 grams of protein and 21 grams of carbohydrates. A typical slice of “lite” rye bread contains 4 grams of protein and 12 grams of carbohydrates. Sara planning a meal of “beans-on-toast” and she want it to supply 20 grams of protein and 80 grams of carbohydrates. How should she prepare her meal? Answer:
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12. METHODS OF SOLVING SYSTEMS OF EQUATIONS
B) Gauss Elimination Consider a system of linear equations: The system can be written in the augmented form [A|B].
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13. Gauss Elimination Method
The augmented form: X=[A|B] By using ERO, A may be reduce in REF/ Upper Triangular form. (Refer Types of Matrices)
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14. Gauss Elimination Method
Example Solve the system by using the Gauss elimination method Answer:
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15. Gauss Elimination Method
Solution
Write in augmented form: X=[A|B]
Apply ERO, use appropriate characteristics
Apply backward substitution
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16. Gauss Elimination Method
Example Solve the system by using the Gauss elimination method Answer:
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17. Gauss Elimination Method
Example You manage an ice cream factory that makes three flavors: Creamy Vanilla, Continental Mocha, and Succulent Strawberry. Into each batch of Creamy Vanilla go two eggs, one cup of milk, and two cups of cream. Into each batch of Continental Mocha go one egg, one cup of milk, and two cups of cream. Into each batch of Succulent Strawberry go one egg, two cups of milk, and one cup of cream. Your stocks of eggs, milk, and cream vary from day to day. How many batches of each flavor should you make in order to use up all of your ingredients if you have 350 eggs, 350 cups of milk and 400 cups of cream?
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18. Gauss Elimination Method
Answer: 100 Creamy Vanilla flavour 50 Continental Mocha flavour 100 Succulent Strawberry flavour
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19. METHODS OF SOLVING SYSTEMS OF EQUATIONS
C) Gauss-Jordan Elimination Consider a system of linear equations: The system can be written in the augmented form [A|B].
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20. Gauss-Jordan Elimination Method
The augmented form: X=[A|B] By using ERO, A may be reduce in RREF/ Identity form. (Refer Types of Matrices)
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21. Gauss-Jordan Elimination Method
Example Solve the system by using the Gauss-Jordan elimination method Answer:
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22. Gauss-Jordan Elimination Method
Solution
Write in augmented form: X=[A|B]
Apply ERO, use appropriate characteristics:
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23. Gauss Elimination Method
Example Solve the system by using the Gauss-Jordan elimination method Answer:
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24. Gauss-Jordan Elimination Method
Example The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day if they have 650 quarts of pineapple juice, 800 quarts of orange juice and 350 quarts of kiwi juice?
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25. Gauss-Jordan Elimination Method
Answer: 100 gallons of PineOrange 150 gallons of PineKiwi 200 gallions of OrangeKiwi
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26. METHODS OF SOLVING SYSTEMS OF EQUATIONS
D) Cramer’s Rule Given the systems of linear equation as: Where With D is the determinant of A (D=det(A)) and
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27. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Cramer’s rule for a 3x3 system: Since Then
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28. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
The solutions for and are given by: Where
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29. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Step of solution
Compute D, determinant of A and
Compute Di , i=1,2,3,…n, where Di is obtained from D by replacing the ith column of D by B
Solution are given by
Cramer’s rule cannot be applies if D=0. In such a case, either a unique solution to the system does not exist or there is no solution.
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30. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Example Solve the system by using the Cramer’s rule method Answer:
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31. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Solution
Write down A, B and X.
Find D (determinant of A)
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32. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Example Solve the system by using the Crames’r rule method Answer:
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33. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Example The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day if they have 800 quarts of pineapple juice, 650 quarts of orange juice and 350 quarts of kiwi juice?
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34. nhaa/imk/sem120162017/eqt101/rk12/32
Cramer’s Rule
Answer: 100 gallons of PineOrange 200 gallons of PineKiwi 150 gallions of OrangeKiwi
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