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Unformatted text preview: 10/4/2018 5.3/3.6 - Matrix Inversion and Determinants John Hardin Fall 2018 MAT211, section 71283, Fall 2018 5.3/3.6 ­ Matrix Inversion and Determinants (Homework) Instructor: Jonathan Perinetti Current Score : 95.42 / 100 Due : Tuesday, September 18 2018 11:59 PM MSTLast Saved : n/a Saving... WebAssign The due date for this assignment is past. Your work can be viewed below, but no changes can be made. Important! Before you view the answer key, decide whether or not you plan to request an extension. Your Instructor may not grant you an extension if you have viewed the answer key. Automatic extensions are not granted if you have viewed the answer key. View Key 1. 8.75/8.33 points | Previous Answers Determine whether or not the given pair of matrices is an inverse pair. NOTE: You will only have ONE attempt on this question. A = 1 1 3 1 −1 0 0 1 3 , B = 0 1 −3 0 0 1 0 0 1 Yes No 2. 8.75/8.33 points | Previous Answers First find the determinant and then find the inverse of the given matrix if it exists. If the inverse does exist, you should check your answer by multiplication. If the inverse doesn't exist, enter DNE in all answer boxes. HINT [See Example 2.] 1 2 A = 1 1 |A|= ­1 A−1 = ­1 2 1 ­1 1/5 10/4/2018 5.3/3.6 - Matrix Inversion and Determinants 3. 8.75/8.33 points | Previous Answers First find the determinant and then find the inverse of the given matrix if it exists. If the inverse does exist, you should check your answer by multiplication. If the inverse doesn't exist, enter DNE in all answer boxes. HINT [See Example 2.] 4 0 B = 0 3 |B|= 12 B−1 = 1/4 0 0 1/3 4. 8.75/8.33 points | Previous Answers First find the determinant and then find the inverse of the given matrix if it exists. If the inverse does exist, you should check your answer by multiplication. If the inverse doesn't exist, enter DNE in all answer boxes. HINT [See Example 2.] C = 6 −1 −18 3 |C|= 0 C−1 = DNE DNE DNE DNE 5. 8.75/8.33 points | Previous Answers Consider the following matrix, A. For what value of k will |A|=28? A = 2 k 2 2 ­12 6. 8.75/8.33 points | Previous Answers Consider the following matrix, A. For what value of p will A−1 not exist? A = 10 p 4 8 20 2/5 10/4/2018 5.3/3.6 - Matrix Inversion and Determinants 7. 0/8.33 points | Previous AnswersWaneFMAC7 5.3.067. If A and B are square matrices with AB = I and BA = I, then which of the following is true? At least one of A and B is singular. A and B must be equal. A and B must both be singular. B is the inverse of A. 8. 8.75/8.33 points | Previous Answers If a system of linear equations is represented in matrix form AX=B, then which of the following is an appropriate equation for solving the system? X = A­1B X = AB­1 X = BA­1 X = B­1 9. 8.75/8.33 points | Previous AnswersWaneFMAC7 5.3.043. Use matrix inversion to solve the given system of linear equations. (You previously solved this system using row reduction in Chapter 4.) x + y = 2 x − y = 1 (x, y) = $$32 ,12 3/5 10/4/2018 5.3/3.6 - Matrix Inversion and Determinants 10.8.75/8.33 points | Previous AnswersWaneFMAC7 5.3.044. Use matrix inversion to solve the given system of linear equations. (You previously solved this system using row reduction in Chapter 4.) −6x + y = 6 −6x − 5y = 6 (x, y) = $$−1,0 11.8.33/8.33 points | Previous AnswersWaneFMAC7 5.3.063. Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes [1 2 15 18 20 0 13 9 19 19 9 15 14]. To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix 1 3 . We can first arrange the coded sequence of numbers in the 4 2 form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A. Encrypted Matrix = = 1 3 1 15 20 13 19 4 2 2 18 7 69 20 40 0 76 9 9 19 15 54 14 8 96 80 70 114 66 56 14 0 , which we can also write as [7 8 69 96 20 80 40 70 76 114 54 66 14 56]. To decipher the encoded message, multiply the encrypted matrix by A−1. The following exercise uses the above matrix A for encoding and decoding. Use the matrix A to encode the phrase "GO TO PLAN B". 52 58 60 40 15 60 52 88 43 32 6 4 4/5 10/4/2018 5.3/3.6 - Matrix Inversion and Determinants 12.8.37/8.37 points | Previous AnswersWaneFMAC7 5.3.065. Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes [1 2 15 18 20 0 13 9 19 19 9 15 14]. To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix 2 4 1 3 . We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A. Encrypted Matrix = = 2 4 1 15 20 13 19 1 3 2 18 0 9 9 19 15 10 102 40 62 114 78 28 7 69 20 40 76 54 14 14 0 , which we can also write as [10 7 102 69 40 20 62 40 114 76 78 54 28 14]. To decipher the encoded message, multiply the encrypted matrix by A−1. The following question uses the above matrix A for encoding and decoding. Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.) [66 48 108 72 22 14 40 20 58 43 130 88 82 59] CORRECT ANSWER 5/5 ...
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