1. Inverse Matrices and Systems

2. 1) Inverse Matrices and Systems of Equations
You have solved systems of equations using graphing, substitution, elimination…oh my…
In the “real world”, these methods take too long and are almost never used.
Inverse matrices are more practical.

3. 1) Inverse Matrices and Systems of Equations
For a
System of Equations

4. 1) Inverse Matrices and Systems of Equations
For a We can write a
System of Equations Matrix Equation

5. 1) Inverse Matrices and Systems of Equations
Example 1:
Write the system as a matrix equation

6. 1) Inverse Matrices and Systems of Equations
Example 1:
Write the system as a matrix equation
Matrix Equation

7. 1) Inverse Matrices and Systems of Equations
Example 1:
Write the system as a matrix equation
Matrix Equation
Coefficient matrix
Variable matrix
Constant matrix

8. 1) Inverse Matrices and Systems of Equations
Example 2:

9. 1) Inverse Matrices and Systems of Equations
Example 2:

10. 1) Inverse Matrices and Systems of Equations
Example 2: A X B

11. 1) Inverse Matrices and Systems of Equations

12. 1) Inverse Matrices and Systems of Equations
When rearranging, take the inverse of A

13. 1) Inverse Matrices and Systems of Equations
The Plan…
“Solve the system” using matrices and inverses

14. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system

15. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 1: Write a matrix equation

16. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 1: Write a matrix equation

17. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 2: Find the determinant and A-1

18. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 2: Find the determinant and A-1
Change signs
Change places

19. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 2: Find the determinant and A-1
Change signs
Change places
detA = 4 – 3
= 1

20. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 2: Find the determinant and A-1

21. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 3: Solve for the variable matrix

22. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 3: Solve for the variable matrix

23. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 3: Solve for the variable matrix

24. 1) Inverse Matrices and Systems of Equations
Example 3:
Solve the system
Step 3: Solve for the variable matrix
The solution to the system is (4, 1).

25. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.

26. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.

27. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.
detA =
= 1

28. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.

29. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.
The solution to the system is (-1, 4).

30. 1) Inverse Matrices and Systems of Equations
Example 4:
Solve the system. Check your answer.
Check

31. 1) Inverse Matrices and Systems of Equations
What about a matrix that has no inverse?
It will have no unique solution.

32. 1) Inverse Matrices and Systems of Equations
Example 5:
Determine whether the system has a unique solution.

33. 1) Inverse Matrices and Systems of Equations
Example 5:
Determine whether the system has a unique solution.
Find the determinant.

34. 1) Inverse Matrices and Systems of Equations
Example 5:
Determine whether the system has a unique solution.
Find the determinant.

35. 1) Inverse Matrices and Systems of Equations
Example 5:
Determine whether the system has a unique solution.
Find the determinant.
Since detA = 0, there is no inverse.
The system does not have a unique solution.

36. Homework
p.217 #1-5, 7-10, 20, 21, 23, 24, 26, 27, 36
DUE TOMORROW: Two codes
TEST: Wednesday Nov 25
Chapter 4