1. Review of Matrices
Or A Fast Introduction

2. Fundamental Tools For Linear Systems
Why Study Matrices?
Fundamental Tools For Linear Systems

3. Matrices Come From Systems of Equations
We’ve known how to find
For different right hand sides

4. Matrices Come From Systems of Equations
Matrix
Vector
Matrices are all about what happens with different

5. Matrices Come From Systems of Equations
Matrices are all about what happens with different

6. Matrices Come From Systems of Equations
Fundamental Definition:
Matrix Vector Multiplication
(Think of it as a system of equations!)
Try it!

7. Matrices Come From Systems of Equations
Fundamental Definition:
Matrix Vector Multiplication
(Think of it as a system of equations!)
Try it!

8. Matrices Come From Systems of Equations
Fundamental Definition:
Matrix Vector Multiplication
Rows
Notation
Columns
Second Index
is Column
First Index
Is Row

9. Matrices Come From Systems of Equations
Fundamental Definition:
Matrix Vector Multiplication
Notation
Second Column
First Row

10. Matrices Come From Systems of Equations
Other Important Definitions
Addition

11. Matrices Come From Systems of Equations
Other Important Definitions
Addition

12. Matrices Come From Systems of Equations
Other Important Definitions
Addition

13. Matrices Come From Systems of Equations
Other Important Definitions
Addition

14. Matrices Come From Systems of Equations
Other Important Definitions
Addition

15. Matrices Come From Systems of Equations
Other Important Definitions
Addition

16. Matrices Come From Systems of Equations
Other Important Definitions
Addition
In General

17. Matrices Come From Systems of Equations
Other Important Definitions
Addition
In General

18. Matrices Come From Systems of Equations
Other Important Definitions
Addition
In General

19. Matrices Come From Systems of Equations
Other Important Definitions
Scalar Multiplication

20. Matrices Come From Systems of Equations
Other Important Definitions
Addition and Scalar Multiplication Give Subtraction
Without a Vector

21. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

22. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

23. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

24. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

25. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

26. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

27. Matrices Come From Systems of Equations
Other Important Definitions
Matrix Multiplication

28. Special Matrices
Identity
Zero

29. Matrix Algebra Rules A Lot Like Normal Algebra Rules
(Addition Commutes)
(Addition Associates)
(Multiplication Associates)
(Distribute)
(Scalars Distribute)
Notice:
(Multiplication
DOES NOT
Commute)

30. Integration and Differentiation

31. Integration and Differentiation

32. Matrix Inversion Many Matrices Have an Inverse
Inverse Has Special Properties
For Example:

33. Inverse Exists If and Only If
Inverse Formula
Matrix Inverse For 2 by 2
The Determinant
Inverse Exists If and Only If

34. Inverse Helps Solve Systems of Equations

35. Inverse Helps Solve Systems of Equations

36. Inverse Helps Solve Systems of Equations

37. Inverse Helps Solve Systems of Equations

38. Inverse Helps Solve Systems of Equations

39. Inverse Helps Solve Systems of Equations

40. The Inverse Does Not Exist!
Singular Matrices
Matrix Inverse For 2 by 2
If the Determinant is 0
The Inverse Does Not Exist!
May Not Have A Solution

41. Example
Determinant is 0
Impossible!

42. If (and only if) the Determinant is 0
Really Important Fact
If (and only if) the Determinant is 0
There is a Vector
Where

43. Example

44. Three Ways of Saying The Same Thing
(1) The Determinant is 0
(2) The Matrix Has No Inverse (Is Singular)
(3) There Is A Vector
Where

45. Summary Matrices Come From Systems Of Equations
We can do Algebra on Matrices
Matrices are called “singular” if the determinant is 0. Such matrices have no inverses.

46. Questions?