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Five-Minute Check (over Lesson 3–7) CCSS Then/Now New Vocabulary Key Concept: Identity Matrix for Multiplication Example 1: Verify Inverse Matrices Key Concept: Inverse of a 2 × 2 Matrix Example 2: Find the Inverse of a Matrix

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  1. Five-Minute Check (over Lesson 3–7) CCSS Then/Now New Vocabulary Key Concept: Identity Matrix for Multiplication Example 1: Verify Inverse Matrices Key Concept: Inverse of a 2 × 2 Matrix Example 2: Find the Inverse of a Matrix Example 3: Real-World Example: Solve a System of Equations Lesson Menu

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

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

  8. 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 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 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 Find the determinant. Since the determinant is not equal to 0, S–1 exists. Example 2

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

  18. Answer: Find the Inverse of a Matrix Definition of inverse a = –1, b = 0, c = 8, d = –2 Simplify. 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 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 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. 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. 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. 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.3x + 4y = –10x – 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.3x + 4y = –10x – 2y = 10 A. (–2, 4) B. (2, –4) C. (–4, 2) D. no solution Example 3

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