Explanation Calculation: Consider the matrix, A = 1 − 1 0 1 0 − 1 6 − 2 − 3 . And the cryptogram is, 110 − 39 − 59 25 − 21 − 3 23 − 18 − 5 47 − 20 − 24 149 − 56 − 75 87 − 38 − 37 . Use the elementary row operations to compute A − 1 as A ⋮ I = I ⋮ A − 1 . Write the augmentation of the matrix A with the identity matrix of order 3 × 3 , A ⋮ I = 1 − 1 0 1 0 − 1 6 − 2 − 3 ⋮ ⋮ ⋮ 1 0 0 0 1 0 0 0 1 Apply the row operations to reduce matrix A to I , Make the elements below the leading element of the first column 0 by multiplying row 1 by − 1 and adding the result to row 2 . Then, multiply row 1 by − 6 and add the result to row 3 as shown, − 1 R 1 + R 2 → − 6 R 1 + R 3 → 1 − 1 0 0 1 − 1 0 4 − 3 ⋮ ⋮ ⋮ 1 0 0 − 1 1 0 − 6 0 1 Add row 2 to row 1 and add − 4 of row 2 to row 3 , R 2 + R 1 → − 4 R 2 + R 3 → 1 0 − 1 0 1 − 1 0 0 1 ⋮ ⋮ ⋮ 0 1 0 − 1 1 0 − 2 − 4 1 Add row 3 to row 1 and row 2 , R 3 + R 1 → R 3 + R 2 → 1 0 0 0 1 0 0 0 1 ⋮ ⋮ ⋮ − 2 − 3 1 − 3 − 3 1 − 2 − 4 1 The above matrix results in I ⋮ A − 1 . Therefore, the inverse of the matrix A is A − 1 = − 2 − 3 1 − 3 − 3 1 − 2 − 4 1 . Now, divide the cryptogram into groups of three to form the coded row matrices, 110 − 39 − 59 25 − 21 − 3 23 − 18 − 5 47 − 20 − 24 149 − 56 − 75 87 − 38 − 37 Obtain the decoded matrices for the coded row matrices by multiplying coded row matrices by the matrix A − 1 = − 2 − 3 1 − 3 − 3 1 − 2 − 4 1

10th Edition

ISBN: 9781337282291

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 7.5, Problem 10ECP

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To calculate: The decoded message of the cryptogram 110−39−5925−21−323−18−547−20−24149−56−7587−38−37 using the inverse of the matrix A=1−1010−16−2−3.

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Chapter 7.5, Problem 9ECP

Chapter 7.5, Problem 1E

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The decoded message of the cryptogram 110 − 39 − 59 25 − 21 − 3 23 − 18 − 5 47 − 20 − 24 149 − 56 − 75 87 − 38 − 37 using the inverse of the matrix A = 1 − 1 0 1 0 − 1 6 − 2 − 3 .

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To calculate: The decoded message of the cryptogram 110−39−5925−21−323−18−547−20−24149−56−7587−38−37 using the inverse of the matrix A=1−1010−16−2−3.

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Chapter 7 Solutions