1. Splash Screen

2. Mathematical Practices 5 Use appropriate tools strategically.
Content Standards
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
Mathematical Practices
5 Use appropriate tools strategically.
CCSS

3. You solved systems of linear equations algebraically.
Find the inverse of a 2 × 2 matrix.
Write and solve matrix equations for a system of equations.
Then/Now

4. identity matrix square matrix inverse matrix matrix equation
variable matrix
constant matrix
Vocabulary

5. Concept

6. A. Determine whether X and Y are inverses.
Verify Inverse Matrices
A. Determine whether X and Y are inverses.
If X and Y are inverses, then X ● Y = Y ● X = I.
Write an equation.
Matrix multiplication
Example 1

7. Matrix multiplication
Verify Inverse Matrices
Write an equation.
Matrix multiplication
Answer:
Example 1

8. Matrix multiplication
Verify Inverse Matrices
Write an equation.
Matrix multiplication
Answer: Since X ● Y = Y ● X = I, X and Y are inverses.
Example 1

9. B. Determine whether P and Q are inverses.
Verify Inverse Matrices
B. Determine whether P and Q are inverses.
If P and Q are inverses, then P ● Q = Q ● P = I.
Write an equation.
Matrix multiplication
Answer:
Example 1

10. B. Determine whether P and Q are inverses.
Verify Inverse Matrices
B. Determine whether P and Q are inverses.
If P and Q are inverses, then P ● Q = Q ● P = I.
Write an equation.
Matrix multiplication
Answer: Since P ● Q  I, they are not inverses.
Example 1

11. A. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
Example 1

12. A. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
Example 1

13. B. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
Example 1

14. B. Determine whether the matrices are inverses.
A. yes
B. no
C. not enough information
D. sometimes
Example 1

15. Concept

16. A. Find the inverse of the matrix, if it exists.
Find the Inverse of a Matrix
A. Find the inverse of the matrix, if it exists.
Find the determinant.
Since the determinant is not equal to 0, S –1 exists.
Example 2

17. Definition of inverse a = –1, b = 0, c = 8, d = –2 Simplify. Answer:
Find the Inverse of a Matrix
Definition of inverse
a = –1, b = 0, c = 8, d = –2
Simplify.
Answer:
Example 2

18. Definition of inverse a = –1, b = 0, c = 8, d = –2 Simplify. Answer:
Find the Inverse of a Matrix
Definition of inverse
a = –1, b = 0, c = 8, d = –2
Simplify.
Answer:
Example 2

19. Find the Inverse of a Matrix
Check Find the product of the matrices. If the product is I, then they are inverse. 
Example 2

20. B. Find the inverse of the matrix, if it exists.
Find the Inverse of a Matrix
B. Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer:
Example 2

21. B. Find the inverse of the matrix, if it exists.
Find the Inverse of a Matrix
B. Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer: Since the determinant equals 0, T –1 does not exist.
Example 2

22. A. Find the inverse of the matrix, if it exists.
A. B.
C. D. No inverse exists.
Example 2

23. A. Find the inverse of the matrix, if it exists.
A. B.
C. D. No inverse exists.
Example 2

24. B. Find the inverse of the matrix, if it exists.
A. B.
C. D. No inverse exists.
Example 2

25. B. Find the inverse of the matrix, if it exists.
A. B.
C. D. No inverse exists.
Example 2

26. A system of equations to represent the situation is as follows.
Solve a System of Equations
RENTAL COSTS The Booster Club for North High School plans a picnic. The rental company charges $15 to rent a popcorn machine and $18 to rent a water cooler. The club spends $261 for a total of 15 items. How many of each do they rent?
A system of equations to represent the situation is as follows.
x + y = 15
15x + 18y = 261
Example 3

27. STEP 1 Find the inverse of the coefficient matrix.
Solve a System of Equations
STEP 1 Find the inverse of the coefficient matrix.
STEP 2 Multiply each side of the matrix equation by the inverse matrix.
Example 3

28. Solve a System of Equations
The solution is (3, 12), where x represents the number of popcorn machines and y represents the number of water coolers.
Answer:
Example 3

29. Answer: The club rents 3 popcorn machines and 12 water coolers.
Solve a System of Equations
The solution is (3, 12), where x represents the number of popcorn machines and y represents the number of water coolers.
Answer: The club rents 3 popcorn machines and 12 water coolers.
Example 3

30. Use a matrix equation to solve the system of equations
Use a matrix equation to solve the system of equations. 3x + 4y = –10 x – 2y = 10
A. (–2, 4)
B. (2, –4)
C. (–4, 2)
D. no solution
Example 3

31. Use a matrix equation to solve the system of equations
Use a matrix equation to solve the system of equations. 3x + 4y = –10 x – 2y = 10
A. (–2, 4)
B. (2, –4)
C. (–4, 2)
D. no solution
Example 3

32. End of the Lesson