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Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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Splash Screen

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Lesson 7 Contents Example 1Verify Inverse Matrices Example 2Find the Inverse of a Matrix Example 3Use Inverses to Solve a Problem

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Example 7-1a Determine whether X and Y are inverses. Check to see if X Y = I. Write an equation. Matrix multiplication

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Example 7-1b Now find Y X. Matrix multiplication Write an equation. Answer: Since X Y = Y X = I, X and Y are inverses.

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Example 7-1c Determine whether P and Q are inverses. Check to see if P Q = I. Write an equation. Matrix multiplication Answer: Since P Q I, they are not inverses.

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Example 7-1d Determine whether each pair of matrices are inverses. a. b. Answer: no Answer: yes

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Example 7-2a Find the inverse of the matrix, if it exists. Find the value of the determinant. Since the determinant is not equal to 0, S –1 exists.

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Example 7-2b Definition of inverse a = –1, b = 0, c = 8, d = –2 Answer:Simplify.

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Example 7-2c Check:

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Example 7-2d 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.

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Find the inverse of each matrix, if it exists. a. b. Example 7-2e Answer: No inverse exists. Answer:

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Example 7-3a Use the table to assign a number to each letter in the message ALWAYS_SMILE. Then code the message with the matrix Code _0_0I 9R18 A1A1J10S19 B2B2K11T20 C3C3L12U21 D4D4M13V22 E5E5N14W23 F6F6O15X24 G7G7P16Y25 H8H8Q17Z26 ALWAYS_SMILE Convert the message to numbers using the table.

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Example 7-3b Write the message in matrix form. Then multiply the message matrix B by the coding matrix A. Write an equation.

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Example 7-3c Matrix multiplication

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Example 7-3d Simplify. Answer: The coded message is 13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39.

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Example 7-3e Use the inverse matrix A –1 to decode 13 | 38 | 24 | 49 | 44 | 107 |19 | 57 | 22 | 53 | 17 | 39. Definition of inverse a = 1, b = 2, c = 1, d = 3 Simplify.

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Example 7-3f Next, decode the message by multiplying the coded matrix C by A –1.

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Example 7-3g

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Example 7-3h

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Example 7-3i Use the table again to convert the numbers to letters. You can now read the message. Code _0_0I 9R18 A1A1J10S19 B2B2K11T20 C3C3L12U21 D4D4M13V22 E5E5N14W23 F6F6O15X24 G7G7P16Y25 H8H8Q17Z26 Answer: ALWAYS_SMILE

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Example 7-3j Code _0_0I 9R18 A1A1J10S19 B2B2K11T20 C3C3L12U21 D4D4M13V22 E5E5N14W23 F6F6O15X24 G7G7P16Y25 H8H8Q17Z26 a.Use the table to assign a number to each letter in the message FUN_MATH. Then code the message with the matrix A = Answer: 12 | 63 | 28 | 14 | 26 | 16 | 40 | 44

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Example 7-3k Code _0_0I 9R18 A1A1J10S19 B2B2K11T20 C3C3L12U21 D4D4M13V22 E5E5N14W23 F6F6O15X24 G7G7P16Y25 H8H8Q17Z26 Answer: b.Use the inverse matrix of to decode the message 12 | 63 | 28 | 14 | 26 | 16 | 40 | FUN_MATH

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