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

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Jason Aubrey

This document is a lecture on identity matrices given by Jason Aubrey of the University of Missouri Department of Mathematics. It defines an identity matrix as an n×n matrix with 1s on the main diagonal and 0s elsewhere. Examples of 2×2 and 3×3 identity matrices are given. The key property that the product of a matrix and the identity matrix equals the original matrix is also described. An example calculation of multiplying two matrices is shown step-by-step to illustrate the use of the identity matrix.

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Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Math 1300 Finite Mathematics Section 4.5 Inverse of a Square Matrix Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (Identity Matrix for Multiplication) An n × n matrix with the properties that university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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 is called the n × n identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1 is the 3 × 3 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1 is the 3 × 3 identity matrix. The reason In is called ’the n × n identity matrix’ is because AIn = A In B = B university-logo whenever those products are defined. Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 2(1) − 1(0) = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 2(1) − 1(0) 2 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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
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
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
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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition Let M be a square matrix of order n and I be the identity matrix of order n. If there exists a matrix M −1 (read "M inverse") such that M −1 M = MM −1 = I then M −1 is called the multiplicative inverse of M or, more simply, the inverse of M. If no such matrix exists, then M is said to be a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 3 −4 3 4 Example: The matrices and are inverses of −2 3 2 3 each other because 3 −4 3 4 1 0 = −2 3 2 3 0 1 and 3 4 3 −4 1 0 = 2 3 −2 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 2 2 1 1 0 0 Example: Since = We conclude −1 −1 −1 −1 0 0 2 2 1 1 that 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To find the inverse of a square matrix M, university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To find the inverse of a square matrix M, 1 Form the augmented matrix [M |I ] university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 2 −6 M= 1 −2 Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 2 −6 M= 1 −2 Find the inverse of M, if it exists. −1 3 M −1 = 1 −2 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find the inverse of M, if it exists. 3 1 1 0 6 2 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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 1 Here we have a problem: the row with zeros on the left indicates that our matrix M has no inverse. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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 1 Here we have a problem: the row with zeros on the left indicates that our matrix M has no inverse. During the process of finding the inverse of M, if a row results with all zeros on the left of the vertical bar (the M side), then M has no inverse. In this case, M is called a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Cryptography Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Cryptography 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
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 university-logo Jason Aubrey Math 1300 Finite Mathematics
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 university-logo Jason Aubrey Math 1300 Finite Mathematics
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 For example, the sequence 19 5 3 18 5 20 0 3 15 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
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 For example, the sequence 19 5 3 18 5 20 0 3 15 4 5 corresponds to the (plaintext) message “SECRET CODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, first we divide the numbers in the sequence into groups of 2 and use these groups as the columns of a matrix B with 2 rows. university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, first we divide the numbers in the sequence into groups of 2 and use these groups as the columns 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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 Thus 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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 Thus the coded message (the ciphertext) is 91 24 66 21 80 25 9 3 72 19 20 5 This message can be decoded by putting it back into matrix form and multiplying on the left by the decoding matrix A−1 university-logo Jason Aubrey Math 1300 Finite Mathematics
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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 To 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
Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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 To 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
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 0 . university-logo Jason Aubrey Math 1300 Finite Mathematics
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 0 This gives the sequence 19 5 3 18 5 20 0 3 15 4 5 . university-logo Jason Aubrey Math 1300 Finite Mathematics
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 0 This gives the sequence 19 5 3 18 5 20 0 3 15 4 5 And this corresponds to the plaintext message “SECRET CODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics

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

  • 1. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Math 1300 Finite Mathematics Section 4.5 Inverse of a Square Matrix Jason Aubrey Department of Mathematics University of Missouri university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 2. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (Identity Matrix for Multiplication) An n × n matrix with the properties that university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 3. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition (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 is called the n × n identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 6. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 7. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1 is the 3 × 3 identity matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 8. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography For example, 1 0 I2 = 0 1 is the 2 × 2 identity matrix.   1 0 0 I3 = 0 1 0 0 0 1 is the 3 × 3 identity matrix. The reason In is called ’the n × n identity matrix’ is because AIn = A In B = B university-logo whenever those products are defined. Jason Aubrey Math 1300 Finite Mathematics
  • 9. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 10. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 11. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 2(1) − 1(0) = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 12. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 2 −1 1 0 2(1) − 1(0) 2 = = 1 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 13. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example:    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. 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. 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. 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Definition Let M be a square matrix of order n and I be the identity matrix of order n. If there exists a matrix M −1 (read "M inverse") such that M −1 M = MM −1 = I then M −1 is called the multiplicative inverse of M or, more simply, the inverse of M. If no such matrix exists, then M is said to be a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 35. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 3 −4 3 4 Example: The matrices and are inverses of −2 3 2 3 each other because 3 −4 3 4 1 0 = −2 3 2 3 0 1 and 3 4 3 −4 1 0 = 2 3 −2 3 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 36. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography 2 2 1 1 0 0 Example: Since = We conclude −1 −1 −1 −1 0 0 2 2 1 1 that 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To find the inverse of a square matrix M, university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 38. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To find the inverse of a square matrix M, 1 Form the augmented matrix [M |I ] university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 39. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography To 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 2 −6 M= 1 −2 Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 42. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 2 −6 M= 1 −2 Find the inverse of M, if it exists. −1 3 M −1 = 1 −2 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 43. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find the inverse of M, if it exists. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 44. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find the inverse of M, if it exists. 3 1 1 0 6 2 0 1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 45. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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 1 Here we have a problem: the row with zeros on the left indicates that our matrix M has no inverse. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 48. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Example: Let 3 1 M= 6 2 Find 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 1 Here we have a problem: the row with zeros on the left indicates that our matrix M has no inverse. During the process of finding the inverse of M, if a row results with all zeros on the left of the vertical bar (the M side), then M has no inverse. In this case, M is called a singular matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 49. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Cryptography Matrix inverses can provide a simple and effective procedure for encoding and decoding messages. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 50. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Cryptography 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. 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 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 52. 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 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 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 26 For example, the sequence 19 5 3 18 5 20 0 3 15 4 5 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 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 26 For example, the sequence 19 5 3 18 5 20 0 3 15 4 5 corresponds to the (plaintext) message “SECRET CODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 55. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 56. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 57. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, first we divide the numbers in the sequence into groups of 2 and use these groups as the columns of a matrix B with 2 rows. university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 58. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Any matrix whose elements are positive integers and whose inverse exists can be used as an encoding matrix. For example, to use the 2 × 2 matrix 4 3 A= 1 1 to encode the preceeding message, first we divide the numbers in the sequence into groups of 2 and use these groups as the columns 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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 Thus 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography Then 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 Thus the coded message (the ciphertext) is 91 24 66 21 80 25 9 3 72 19 20 5 This message can be decoded by putting it back into matrix form and multiplying on the left by the decoding matrix A−1 university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 63. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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 To 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. Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography We 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 To 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. 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 0 . university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 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 0 This gives the sequence 19 5 3 18 5 20 0 3 15 4 5 . university-logo Jason Aubrey Math 1300 Finite Mathematics
  • 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 0 This gives the sequence 19 5 3 18 5 20 0 3 15 4 5 And this corresponds to the plaintext message “SECRET CODE”. university-logo Jason Aubrey Math 1300 Finite Mathematics