1. TYPES OF SOLUTIONS SOLVING EQUATIONS
CHAPTER 2 MATRICES
TYPES OF SOLUTIONS
SOLVING EQUATIONS

2. TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
There are 3 possible solutions:
3 TYPES OF SOLUTIONS
A SYSTEM WITH UNIQUE SOLUTION
A SYSTEM WITH INFINITELY MANY SOLUTIONS
A SYSTEM WITH NO SOLUTION
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3. A SYSTEMS WITH UNIQUE SOLUTION
Consider the system:
Augmented matrix:
The system has unique solution:
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4. A SYSTEMS WITH INFINITELY MANY SOLUTION
Consider the system:
Augmented matrix:
The system has many solutions: let where s is called a free variable. Then,
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5. A SYSTEMS WITH NO SOLUTION
Consider the system:
Augmented matrix:
The system has no solution, since coefficient of is ‘0’.
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6. SOLVING SYSTEMS OF EQUATIONS
Systems of linear equations :
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7. SOLVING SYSTEMS OF EQUATIONS
4 methods used to solve systems of equations.
The Inverse of the Coefficient Matrix
Gauss Elimination
Gauss-Jordan Elimination
Cramer’s Rule
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8. SOLVING SYSTEMS OF EQUATIONS
Matrix Form: AX = B
To find X: X =A-1 B
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9. THE INVERSE OF THE COEFFICIENT MATRIX
Method : X =A-1 B
Example:
Solve the system by using A-1 , the inverse of the coefficient matrix:
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10. THE INVERSE OF THE COEFFICIENT MATRIX
Solution:
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11. THE INVERSE OF THE COEFFICIENT MATRIX
Find A-1 : Cofactor of A : Therefore:
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12. THE INVERSE OF THE COEFFICIENT MATRIX
Find X :
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13. THE INVERSE OF THE COEFFICIENT MATRIX
Example 2: Solve the system by using A-1 , the inverse of the coefficient matrix: Answer :
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14. GAUSS ELIMINATION
Consider the systems of linear eq:
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15. GAUSS ELIMINATION Write in augmented form : [A|B]
Using ERO, such that A may be reduce in REF/Upper Triangular
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16. GAUSS ELIMINATION
Example: Solve the system by using Gauss Elimination method:
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17. GAUSS ELIMINATION
Solution: Write in augmented form:
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18. Reduce to REF : (Diagonal = 1)
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19. x y z
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20. GAUSS JORDAN ELIMINATION
Written in augmented form : [A|B]
Using ERO, such that A may be reduce in RREF/IDENTITY (DIAGONAL = 1, OTHER ENTRIES = 0)
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21. Reduce to RREF : (Diagonal = 1, Other entries = 0)
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22. NHAA/IMK/UNIMAP

23. Example 2: Solve the system by Gauss elimination. Answer :
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24. Gauss jordan elimination
Example 4: Solve the system by Gauss Jordan elimination. Answer :
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25. CRAMER’S RULE
Theorem 5 Cramer’s Rule for 3x3 system Given the system: with :
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26. CRAMER’S RULE
If : Then :
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27. CRAMER’S RULE
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28. CRAMER’S RULE Example 5: Solve the system by using the Cramer’s Rule.
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29. CRAMER’S RULE
Solution Determinant of A :
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30. CRAMER’S RULE
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31. CRAMER’S RULE
Example 6: Solve the system by using Cramer’s Rule. Answer :
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