Lesson Menu 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

CCSS 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.

Then/Now 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.

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

Concept

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

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

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

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

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

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

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

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

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

Concept

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

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

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

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

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

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

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

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

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

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

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

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

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

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