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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
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
identity matrix square matrix inverse matrix matrix equation variable matrix constant matrix Vocabulary
Concept
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
Matrix multiplication Verify Inverse Matrices Write an equation. Matrix multiplication Answer: Example 1
Matrix multiplication Verify Inverse Matrices Write an equation. Matrix multiplication Answer: Since X ● Y = Y ● X = I, X and Y are inverses. Example 1
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
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
A. Determine whether the matrices are inverses. A. yes B. no C. not enough information D. sometimes Example 1
A. Determine whether the matrices are inverses. A. yes B. no C. not enough information D. sometimes Example 1
B. Determine whether the matrices are inverses. A. yes B. no C. not enough information D. sometimes Example 1
B. Determine whether the matrices are inverses. A. yes B. no C. not enough information D. sometimes Example 1
Concept
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
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
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
Find the Inverse of a Matrix Check Find the product of the matrices. If the product is I, then they are inverse. Example 2
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
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
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 2
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
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
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
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
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
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
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