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Identity Matrix for Multiplication    Inverse of a Square Matrix     Application: CryptographyMath 1300 Finite Mathematics...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication             Inverse of a Square Matrix              Application: CryptographyFor exampl...
Identity Matrix for Multiplication             Inverse of a Square Matrix              Application: CryptographyFor exampl...
Identity Matrix for Multiplication             Inverse of a Square Matrix              Application: CryptographyFor exampl...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication             Inverse of a Square Matrix              Application: CryptographyExample: 2...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication              Inverse of a Square Matrix               Application: CryptographyExample:...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: CryptographyExampl...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: CryptographyExampl...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: CryptographyExampl...
Identity Matrix for Multiplication           Inverse of a Square Matrix            Application: Cryptography           ...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: Cryptography     ...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: Cryptography     ...
Identity Matrix for Multiplication                 Inverse of a Square Matrix                  Application: CryptographyDe...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: Cryptography      ...
Identity Matrix for Multiplication                 Inverse of a Square Matrix                  Application: Cryptography  ...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyTo fi...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyTo fi...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                   Inverse of a Square Matrix                    Application: Cryptograp...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyBlan...
Identity Matrix for Multiplication           Inverse of a Square Matrix            Application: CryptographyBlank A B C D ...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyAny ...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyAny ...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyAny ...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyAny ...
Identity Matrix for Multiplication                 Inverse of a Square Matrix                  Application: CryptographyTh...
Identity Matrix for Multiplication                 Inverse of a Square Matrix                  Application: CryptographyTh...
Identity Matrix for Multiplication                   Inverse of a Square Matrix                    Application: Cryptograp...
Identity Matrix for Multiplication                   Inverse of a Square Matrix                    Application: Cryptograp...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: CryptographyWe hav...
Identity Matrix for Multiplication               Inverse of a Square Matrix                Application: CryptographyWe hav...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyWe h...
Identity Matrix for Multiplication                Inverse of a Square Matrix                 Application: CryptographyWe h...
Identity Matrix for Multiplication           Inverse of a Square Matrix            Application: CryptographyBlank A B C D ...
Identity Matrix for Multiplication                   Inverse of a Square Matrix                    Application: Cryptograp...
Identity Matrix for Multiplication                  Inverse of a Square Matrix                   Application: Cryptography...
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Math 1300: Section 4-5 Inverse of a Square Matrix

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Transcript of "Math 1300: Section 4-5 Inverse of a Square Matrix"

  1. 1. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyMath 1300 Finite MathematicsSection 4.5 Inverse of a Square Matrix Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
  2. 2. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyDefinition (Identity Matrix for Multiplication)An n × n matrix with the properties that university-logo Jason Aubrey Math 1300 Finite Mathematics
  3. 3. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyDefinition (Identity Matrix for Multiplication)An n × n matrix with the properties that every element on the principal diagonal is a 1, and university-logo Jason Aubrey Math 1300 Finite Mathematics
  4. 4. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyDefinition (Identity Matrix for Multiplication)An n × n matrix with the properties that every element on the principal diagonal is a 1, and every other element is 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  5. 5. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyDefinition (Identity Matrix for Multiplication)An n × n matrix with the properties that every element on the principal diagonal is a 1, and every other element is 0is called the n × n identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  6. 6. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyFor example, 1 0 I2 = 0 1is the 2 × 2 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  7. 7. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyFor example, 1 0 I2 = 0 1is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1is the 3 × 3 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  8. 8. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyFor example, 1 0 I2 = 0 1is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1is the 3 × 3 identity matrix.The reason In is called ’the n × n identity matrix’ is because AIn = A In B = B university-logowhenever those products are defined. Jason Aubrey Math 1300 Finite Mathematics
  9. 9. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  10. 10. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  11. 11. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  12. 12. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  13. 13. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  14. 14. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  15. 15. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) university-logo Jason Aubrey Math 1300 Finite Mathematics
  16. 16. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  17. 17. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  18. 18. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  19. 19. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 = = 0 1 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  20. 20. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) = = 0 1 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  21. 21. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 2 = = 0 1 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  22. 22. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 = = 0 1 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  23. 23. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 −1 = = 0 1 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  24. 24. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 −1 = = 0 1 1 3 0(2) + 1(1) university-logo Jason Aubrey Math 1300 Finite Mathematics
  25. 25. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 −1 = = 0 1 1 3 0(2) + 1(1) 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  26. 26. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 −1 = = 0 1 1 3 0(2) + 1(1) 0(−1) + 1(3) 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  27. 27. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: 2 −1 1 0 2(1) − 1(0) 2(0) − 1(1) 2 −1 = = 1 3 0 1 1(1) + 3(0) 1(0) + 3(1) 1 3 1 0 2 −1 2(1) + 1(0) 1(−1) + 3(0) 2 −1 = = 0 1 1 3 0(2) + 1(1) 0(−1) + 1(3) 1 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  28. 28. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample:    1 0 0 2 0 2 0 1 0 −1 1 −3 0 0 1 1 0 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  29. 29. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample:    1 0 0 2 0 2 0 1 0 −1 1 −3 0 0 1 1 0 3   1(2) + 0(−1) + 0(1) 1(0) + 0(1) + 0(0) 1(2) + 0(−3) + 0(3)= 0(2) + 1(−1) + 0(1) 0(0) + 1(1) + 0(0) 0(2) + 1(−3) + 0(3) 0(2) + 0(−1) + 1(1) 0(0) + 0(1) + 1(0) 0(2) + 0(−3) + 1(3) university-logo Jason Aubrey Math 1300 Finite Mathematics
  30. 30. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample:    1 0 0 2 0 2 0 1 0 −1 1 −3 0 0 1 1 0 3   1(2) + 0(−1) + 0(1) 1(0) + 0(1) + 0(0) 1(2) + 0(−3) + 0(3)= 0(2) + 1(−1) + 0(1) 0(0) + 1(1) + 0(0) 0(2) + 1(−3) + 0(3) 0(2) + 0(−1) + 1(1) 0(0) + 0(1) + 1(0) 0(2) + 0(−3) + 1(3)   2 0 2 = −1 1 −3 1 0 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  31. 31. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography   2 0 2 1 0 0−1 1 −3 0 1 0 1 0 3 0 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  32. 32. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography    2 0 2 1 0 0 −1 1 −3 0 1 0 1 0 3 0 0 1   2(1) + 0(0) + 2(0) 2(0) + 0(1) + 2(0) 2(0) + 0(0) + 2(1)= −1(1) + 1(0) − 3(0) −1(0) + 1(1) − 3(0) −1(0) + 1(0) − 3(1) 1(1) + 0(0) + 3(0) 1(0) + 0(1) + 3(0) 1(0) + 0(0) + 3(1) university-logo Jason Aubrey Math 1300 Finite Mathematics
  33. 33. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography    2 0 2 1 0 0 −1 1 −3 0 1 0 1 0 3 0 0 1   2(1) + 0(0) + 2(0) 2(0) + 0(1) + 2(0) 2(0) + 0(0) + 2(1)= −1(1) + 1(0) − 3(0) −1(0) + 1(1) − 3(0) −1(0) + 1(0) − 3(1) 1(1) + 0(0) + 3(0) 1(0) + 0(1) + 3(0) 1(0) + 0(0) + 3(1)   2 0 2 = −1 1 −3 1 0 3 university-logo Jason Aubrey Math 1300 Finite Mathematics
  34. 34. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyDefinitionLet M be a square matrix of order n and I be the identity matrixof order n. If there exists a matrix M −1 (read "M inverse") suchthat M −1 M = MM −1 = Ithen M −1 is called the multiplicative inverse of M or, moresimply, the inverse of M. If no such matrix exists, then M is saidto be a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  35. 35. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 3 −4 3 4Example: The matrices and are inverses of −2 3 2 3each other because 3 −4 3 4 1 0 = −2 3 2 3 0 1and 3 4 3 −4 1 0 = 2 3 −2 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  36. 36. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 2 2 1 1 0 0Example: Since = We conclude −1 −1 −1 −1 0 0 2 2 1 1that and are not inverses of each other. −1 −1 −1 −1(In fact, these matrices have no inverses). university-logo Jason Aubrey Math 1300 Finite Mathematics
  37. 37. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyTo find the inverse of a square matrix M, university-logo Jason Aubrey Math 1300 Finite Mathematics
  38. 38. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyTo find the inverse of a square matrix M, 1 Form the augmented matrix [M |I ] university-logo Jason Aubrey Math 1300 Finite Mathematics
  39. 39. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyTo find the inverse of a square matrix M, 1 Form the augmented matrix [M |I ] 2 Use row operations to transform [M |I ] into [I |B ] university-logo Jason Aubrey Math 1300 Finite Mathematics
  40. 40. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyTo find the inverse of a square matrix M, 1 Form the augmented matrix [M |I ] 2 Use row operations to transform [M |I ] into [I |B ] 3 The matrix B is the inverse of M; in other words, M −1 = B university-logo Jason Aubrey Math 1300 Finite Mathematics
  41. 41. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 2 −6 M= 1 −2Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
  42. 42. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 2 −6 M= 1 −2Find the inverse of M, if it exists. −1 3 M −1 = 1 −2 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  43. 43. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
  44. 44. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. 3 1 1 0 6 2 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  45. 45. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. 3 1 1 0 −2R +R →R − − 1− − −2 − − −2 − → 6 2 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  46. 46. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. 3 1 1 0 −2R +R →R 3 1 1 0 − − 1− − −2 − − −2 − → 6 2 0 1 0 0 −2 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  47. 47. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. 3 1 1 0 −2R1 +R2 →R2 3 1 1 0 −− − − − − − − −→ 6 2 0 1 0 0 −2 1Here we have a problem: the row with zeros on the leftindicates that our matrix M has no inverse. university-logo Jason Aubrey Math 1300 Finite Mathematics
  48. 48. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyExample: Let 3 1 M= 6 2Find the inverse of M, if it exists. 3 1 1 0 −2R1 +R2 →R2 3 1 1 0 −− − − − − − − −→ 6 2 0 1 0 0 −2 1Here we have a problem: the row with zeros on the leftindicates that our matrix M has no inverse.During the process of finding the inverse of M, if a row resultswith all zeros on the left of the vertical bar (the M side), then Mhas no inverse. In this case, M is called a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  49. 49. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyCryptography Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. university-logo Jason Aubrey Math 1300 Finite Mathematics
  50. 50. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyCryptography Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. To begin, assign the numbers 1-26 to the letters in the alphabet. Also assign the number 0 to a blank to provide for space between words. university-logo Jason Aubrey Math 1300 Finite Mathematics
  51. 51. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyBlank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 university-logo Jason Aubrey Math 1300 Finite Mathematics
  52. 52. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyBlank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13N O P Q R S T U V W X Y Z14 15 16 17 18 19 20 21 22 23 24 25 26 university-logo Jason Aubrey Math 1300 Finite Mathematics
  53. 53. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Blank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26For example, the sequence 19 5 3 18 5 20 0 3 15 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  54. 54. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Blank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26For example, the sequence 19 5 3 18 5 20 0 3 15 4 5corresponds to the (plaintext) message “SECRET CODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics
  55. 55. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyAny matrix whose elements are positive integers and whoseinverse exists can be used as an encoding matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  56. 56. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyAny matrix whose elements are positive integers and whoseinverse exists can be used as an encoding matrix. Forexample, to use the 2 × 2 matrix 4 3 A= 1 1to encode the preceeding message, university-logo Jason Aubrey Math 1300 Finite Mathematics
  57. 57. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyAny matrix whose elements are positive integers and whoseinverse exists can be used as an encoding matrix. Forexample, to use the 2 × 2 matrix 4 3 A= 1 1to encode the preceeding message, first we divide the numbersin the sequence into groups of 2 and use these groups as thecolumns of a matrix B with 2 rows. university-logo Jason Aubrey Math 1300 Finite Mathematics
  58. 58. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyAny matrix whose elements are positive integers and whoseinverse exists can be used as an encoding matrix. Forexample, to use the 2 × 2 matrix 4 3 A= 1 1to encode the preceeding message, first we divide the numbersin the sequence into groups of 2 and use these groups as thecolumns of a matrix B with 2 rows. 19 3 5 0 15 5 B= 5 18 20 3 4 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  59. 59. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyThen we multiply on the left by A: 4 3 19 3 5 0 15 5 AB = 1 1 5 18 20 3 4 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  60. 60. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyThen we multiply on the left by A: 4 3 19 3 5 0 15 5 AB = 1 1 5 18 20 3 4 0 91 66 80 9 72 20 = 24 21 25 3 19 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  61. 61. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyThen we multiply on the left by A: 4 3 19 3 5 0 15 5 AB = 1 1 5 18 20 3 4 0 91 66 80 9 72 20 = 24 21 25 3 19 5Thus the coded message (the ciphertext) is 91 24 66 21 80 25 9 3 72 19 20 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  62. 62. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyThen we multiply on the left by A: 4 3 19 3 5 0 15 5 AB = 1 1 5 18 20 3 4 0 91 66 80 9 72 20 = 24 21 25 3 19 5Thus the coded message (the ciphertext) is 91 24 66 21 80 25 9 3 72 19 20 5This message can be decoded by putting it back into matrixform and multiplying on the left by the decoding matrix A−1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  63. 63. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyWe have 91 66 80 9 72 20 4 3 C= and A = 24 21 25 3 19 5 1 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  64. 64. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyWe have 91 66 80 9 72 20 4 3 C= and A = 24 21 25 3 19 5 1 1 1 −3 A−1 = −1 4 university-logo Jason Aubrey Math 1300 Finite Mathematics
  65. 65. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyWe have 91 66 80 9 72 20 4 3 C= and A = 24 21 25 3 19 5 1 1 1 −3 A−1 = −1 4To decipher the ciphertext, we multiply: 1 −3 91 66 80 9 72 20 A−1 C = −1 4 24 21 25 3 19 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  66. 66. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyWe have 91 66 80 9 72 20 4 3 C= and A = 24 21 25 3 19 5 1 1 1 −3 A−1 = −1 4To decipher the ciphertext, we multiply: 1 −3 91 66 80 9 72 20 A−1 C = −1 4 24 21 25 3 19 5 19 3 5 0 15 5 = 5 18 20 3 4 0 university-logo Jason Aubrey Math 1300 Finite Mathematics
  67. 67. Identity Matrix for Multiplication Inverse of a Square Matrix Application: CryptographyBlank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13N O P Q R S T U V W X Y Z14 15 16 17 18 19 20 21 22 23 24 25 26 19 3 5 0 15 5 P= 5 18 20 3 4 0 . university-logo Jason Aubrey Math 1300 Finite Mathematics
  68. 68. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Blank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26 19 3 5 0 15 5 P= 5 18 20 3 4 0This gives the sequence 19 5 3 18 5 20 0 3 15 4 5 . university-logo Jason Aubrey Math 1300 Finite Mathematics
  69. 69. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Blank A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26 19 3 5 0 15 5 P= 5 18 20 3 4 0This gives the sequence 19 5 3 18 5 20 0 3 15 4 5And this corresponds to the plaintext message “SECRETCODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics
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