1. Chapter 9 Matrices and Determinants Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 9.4 Multiplicative Inverses of Matrices and Matrix Equations

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Find the multiplicative inverse of a square matrix. Use inverses to solve matrix equations. Encode and decode messages. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Multiplicative Identity Matrix The square matrix with 1’s down the main diagonal from upper left to lower right and 0’s elsewhere is called the multiplicative identity matrix of order n. This matrix is designated by I n. For example,

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Definition of the Multiplicative Inverse of a Matrix If a square matrix has a multiplicative inverse, it is said to be invertible or nonsingular. If a square matrix has no multiplicative inverse, it is called singular.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: The Multiplicative Inverse of a Matrix Show that B is the multiplicative inverse of A, where AB = BA = I. Thus, B is the multiplicative inverse of A.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 A Quick Method for Finding the Multiplicative Inverse of a 2 x 2 Matrix

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Using the Quick Method to Find a Multiplicative Inverse Find the multiplicative inverse of ab dc

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Procedure for Finding the Multiplicative Inverse of an Invertible Matrix

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix Find the multiplicative inverse of Step 1 Form the augmented matrix

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Find the multiplicative inverse of Step 2 Perform row operations on to obtain a matrix of the form replace row 2 by

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Step 2 (cont) replace row 3 by

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Step 2 (cont) replace row 2 by replace row 3 by

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Step 2 (cont) replace row 1 by replace row 2 by

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Step 2 Perform row operations on to obtain a matrix of the form The result is Step 3 Matrix B is A –1

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued) Step 4 Verify the result by showing that AA –1 = I 3 and A –1 A = I 3.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Summary: Finding Multiplicative Inverses for Invertible Matrices

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Solving Systems of Equations Using Multiplicative Inverses of Matrices The matrix equation is abbreviated AX = B, where A is the coefficient matrix of the system and X and B are matrices containing one column, called column matrices. The matrix B is called the constant matrix.

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Solving a System Using A –1 If AX = B has a unique solution, then X = A –1 B. To solve a linear system of equations, multiply A –1 and B to find X.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Using the Inverse of a Matrix to Solve a System Solve the system by using A –1, the inverse of the coefficient matrix that was found in the previous example. The linear system can be written as The solution is given by

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Using the Inverse of a Matrix to Solve a System (continued) Solve the system by using A –1, the inverse of the coefficient matrix that was found in the previous example. The solution set is {(4, –2, 1)}.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Applications of Matrix Inverses to Coding A cryptogram is a message written so that no one other than the intended recipient can understand it. To encode a message, we begin by assigning a number to each letter in the alphabet: A = 1, B = 2, C = 3,..., Z = 26, and a space = 0. The numerical equivalent of the message is then converted into a matrix. An invertible matrix can be used to convert the message into code. The multiplicative inverse of this matrix can be used to decode the message.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Encoding a Word or Message

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Encoding a Word Use the coding matrix to encode the word BASE. Step 1 Express the word numerically. The numerical equivalent of BASE is 2,1,19,5. Step 2 List the numbers in step 1 by columns and form a square matrix.

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Encoding a Word (continued) Use the coding matrix to encode the word BASE. Step 3 Multiply the matrix in step 2 by the coding matrix.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Encoding a Word (continued) Use the coding matrix to encode the word BASE. Step 3 Multiply the matrix in step 2 by the coding matrix. The result is Step 4 Use the numbers by columns, from the coded matrix in step 3 to write the encoded message. The encoded message is –7, 10, –53, 77.

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Decoding a Word or Message That Was Encoded

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Decoding a Word Decode –7, 10, –53, 77. Step 1 Find the inverse of the coding matrix. ab cd

28. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Decoding a Word (continued) Decode –7, 10, –53, 77. Step 2 Multiply the multiplicative inverse of the coding matrix and the coded matrix.

29. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 Example: Decoding a Word (continued) Decode –7, 10, –53, 77. Step 3 Express the numbers, by columns, from the matrix in step 2 as letters. The numbers are 2, 1, 19, 5. The decoded message is BASE.