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HW: Pg. 219 #16-26e, 31, 33

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HW: Pg. 219-220 #37, 41, 45, 49, 59

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HW: Quiz 1 Pg. 221 #7-17o

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4.4 Identity and Inverse Matrices

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EXAMPLE 1 Find the inverse of a 2 X 2 matrix Find the inverse of A =. 3 8 2 5

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GUIDED PRACTICE for Example 1 Find the inverse of the matrix. 1. 6 1 2 4 2 11 1 3 – 1 22 – ANSWER 2.–1 5 –4 8 2 3 1 3 5 12 – 1 – ANSWER 3.–3 –4 –1 –2 –1 1 2 2 3 2 – ANSWER

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EXAMPLE 2 Solve a matrix equation SOLUTION Begin by finding the inverse of A. 4 7 1 2 = Solve the matrix equation AX = B for the 2 × 2 matrix X. 2 –7 –1 4 –21 3 12 –2 A B X = A –1 = 1 8 – 7 4 7 1 2

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EXAMPLE 2 Solve a matrix equation To solve the equation for X, multiply both sides of the equation by A – 1 on the left. A –1 AX = A –1 B IX = A –1 B X = A –1 B X = 0 –2 3 –1 4 7 1 2 –21 3 12 –2 = 2 –7 –1 4 4 7 1 2 X X 1 0 0 1 0 –2 3 –1 =

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GUIDED PRACTICE for Example 2 4. Solve the matrix equation –4 1 0 6 X = 8 9 24 6 –1 –2 4 1 ANSWER

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EXAMPLE 3 Find the inverse of a 3 × 3 matrix Use a graphing calculator to find the inverse of A. Then use the calculator to verify your result. 2 1 – 2 5 3 0 4 3 8 A = SOLUTION Enter matrix A into a graphing calculator and calculate A –1. Then compute AA –1 and A –1 A to verify that you obtain the 3 × 3 identity matrix.

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GUIDED PRACTICE for Example 3 5.5. 2 –2 0 2 0 –2 12 –4 –6 A = Use a graphing calculator to find the inverse of the matrix A. Check the result by showing that AA -1 = I and A -1 A = I.

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GUIDED PRACTICE for Example 3 6.6. – 3 4 5 1 5 0 5 2 2 A = 7.7. 2 1 – 2 5 3 0 4 3 8 A =

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EXAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x – 3y = 19 x + 4y = –7 Equation 1 Equation 2 SOLUTION STEP 1 Write the linear system as a matrix equation AX = B. coefficient matrix of matrix of matrix (A) (X) variables constants (B) 2 –3 1 4. xyxy 19 –7 =

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EXAMPLE 4 Solve a linear system STEP 2 Find the inverse of matrix A. 4 3 –1 2 = A –1 = 1 8 – (–3) 4 11 1 3 2 – STEP 3 Multiply the matrix of constants by A –1 on the left. X = A –1 B = 4 11 1 3 – 2 19 –7 = 5 –3 = xyxy The solution of the system is (5, – 3). ANSWER CHECK 2(5) – 3(–3) = 10 + 9 = 19 5 + 4(–3) = 5 – 12 = –7

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EXAMPLE 5 Solve a multi-step problem Gifts A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $15.50. A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $72.50. Find the cost of each item in the gift baskets.

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EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Write verbal models for the situation.

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EXAMPLE 5 Solve a multi-step problem STEP 2 Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 2m + p = 15.50 Equation 1 2m + 2p + d = 37.00 Equation 2 4m + 3p + 2d = 72.50 Equation 3 STEP 3 Rewrite the system as a matrix equation. 2 1 0 2 2 1 4 3 2 mpdmpd 15.50 37.00 72.50 =

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EXAMPLE 5 Solve a multi-step problem STEP 4 Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X = A –1 B. A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.

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GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the linear system. 4x + y = 10 3x + 5y = –1 8.8. (3, –2) ANSWER 9.9. 2x – y = – 6 6x – 3y = – 18 infinitely many solutions ANSWER 10. 3x – y = –5 –4x + 2y = 8 (– 1, 2) ANSWER 11. What if? In Example 5, how does the answer change if a basic basket costs $17, a medium basket costs $35, and a super basket costs $69 ? movie pass: $8 package of popcorn: $1 DVD: $17 ANSWER

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Homework: Pg. 227 #17-23o, 28, 30, 31, 33-36

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