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

3. A SYSTEMS WITH UNIQUE SOLUTION
Consider the system:
Augmented matrix:
The system has unique solution:

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,

5. A SYSTEMS WITH NO SOLUTION
Consider the system:
Augmented matrix:
The system has no solution, since coefficient of is ‘0’.

6. SOLVING SYSTEMS OF EQUATIONS
Systems of linear equations :

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

8. SOLVING SYSTEMS OF EQUATIONS
Matrix Form: AX = B
To find X: X =A-1 B

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:

10. THE INVERSE OF THE COEFFICIENT MATRIX
Solution:

11. THE INVERSE OF THE COEFFICIENT MATRIX
Find A-1 : Cofactor of A : Therefore:

12. THE INVERSE OF THE COEFFICIENT MATRIX
Find X :

13. THE INVERSE OF THE COEFFICIENT MATRIX
Example 2: Solve the system by using A-1 , the inverse of the coefficient matrix: Answer :

14. GAUSS ELIMINATION
Consider the systems of linear eq:

15. GAUSS ELIMINATION Write in augmented form : [A|B]
Using ERO, such that A may be reduce in REF/Upper Triangular

16. GAUSS ELIMINATION
Example: Solve the system by using Gauss Elimination method:

17. GAUSS ELIMINATION
Solution: Write in augmented form:

18. Reduce to REF : (Diagonal = 1)

19. x y z

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)

21. Reduce to RREF : (Diagonal = 1, Other entries = 0)

23. Example 2: Solve the system by Gauss elimination. Answer :

24. Gauss jordan elimination
Example 4: Solve the system by Gauss Jordan elimination. Answer :

25. CRAMER’S RULE
Theorem 5 Cramer’s Rule for 3x3 system Given the system: with :

26. CRAMER’S RULE
If : Then :

27. CRAMER’S RULE

28. CRAMER’S RULE
Example 5: Solve the system by using the Cramer’s Rule.

29. CRAMER’S RULE
Solution Determinant of A :

30. CRAMER’S RULE

31. CRAMER’S RULE
Example 6: Solve the system by using Cramer’s Rule. Answer :