4.7 Identity and Inverse Matrices

1. 4.7 Identity and Inverse Matrices Identity matrices Inverse matrix (intro) An application Finding inverse matrices (by hand) Finding inverse matrices (using calculator)

2. A review of the Identity • For real numbers, what is the additive identity? • Zero…. Why? • Because for any real number b, 0 + b = b • What is the multiplicative identity? • 1 … Why? • Because for any real number b, 1 * b = b

3. Identity Matrices • The identity matrix is a square matrix (same # of rows and columns) that, when multiplied by another matrix, equals that same matrix • If A is any n x n matrix and I is the n x n Identity matrix, then A * I = A and I*A = A

4. Examples • The 2 x 2 Identity matrix is: • The 3 x 3 Identity matrix is: • Notice any pattern? • Most of the elements are 0, except those in the diagonal from upper left to lower right, in which every element is 1!

5. Inverse review • Recall that we defined the inverse of a real number b to be a real number a such that a and b combined to form the identity • For example, 3 and -3 are additive inverses since 3 + -3 = 0, the additive identity • Also, -2 and – ½ are multiplicative inverses since (-2) *(- ½ ) = 1, the multiplicative identity

6. Matrix Inverses • Two n x n matrices are inverses of each other if their product is the identity • Not all matrices have inverses (more on this later) • Often we symbolize the inverse of a matrix by writing it with an exponent of (-1) • For example, the inverse of matrix A is A-1 • A * A-1 = I, the identity matrix.. Also A-1 *A = I • To determine if 2 matrices are inverses, multiply them and see if the result is the Identity matrix!

7. Determine whether X and Y are inverses. Write an equation. Matrixmultiplication Example 7-1a Check to see if X • Y = I.

8. Write an equation. Matrixmultiplication Example 7-1b Now find Y • X. Answer: Since X • Y = Y • X = I, X and Y are inverses.

9. Determine whether P and Q are inverses. Write anequation. Matrix multiplication Example 7-1c Check to see if P • Q = I. Answer: Since P • Q  I,they are not inverses.

10. Determine whether each pair of matrices are inverses. a. b. Example 7-1d Answer: no Answer: yes

11. An Application of Inverse Matrices • You can use matrices to encode and decode a message • In other words, matrices are useful for encrypting information • First, translate your message into numbers using the key A = 1, B = 2, etc.. (perhaps 0 = space) • Organize your message into a matrix with 2 columns and as many rows as needed • Multiply the matrix by a 2 x 2 encoding matrix • To decipher the message, multiply the coded message by a 2 x 2 decoding matrix • The decoding matrix will be the inverse of the encoding matrix • Finally, you can translate the numbers back into letters using you’re the key mentioned above

12. Use the table to assign a number to each letter in the message ALWAYS_SMILE. Then code the message with the matrix Example 7-3a Convert the message to numbers using the table.

13. Write an equation. Example 7-3b Write the message in matrix form. Then multiply the message matrix B by the coding matrix A.

14. Matrix multiplication Example 7-3c

15. Simplify. Example 7-3d Answer: The coded message is 13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39.

16. Now decode the message 13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39 • Decode by: • expressing the coded message as a matrix with 2 columns • Multiplying this matrix by the inverse of A • The inverse of A is shown below:

17. Example 7-3f Next, decode the message by multiplying the coded matrix C by A–1.

18. Example 7-3g

19. Example 7-3h

20. Example 7-3i Use the table again to convert the numbers to letters. You can now read the message. Answer:

21. a. Use the table to assign a number to each letter in the message FUN_MATH. Then code the message with the matrix A = Example 7-3j Answer:12 | 63 | 28 | 14 | 26 | 16 | 40 | 44

22. Example 7-3k Use the inverse matrix shown below to decode the message!! Answer:

23. How do we find the inverse??? • 1st you find what is called the determinant • The determinant not only determines whether the inverse of a matrix exists, but also influences what elements the inverse contains • For the matrix shown below, the determinant is equal to ad – bc • In other words, multiply the elements in each diagonal, then subtract the products!

24. More about determinants • If the determinant of a matrix equals zero, then the inverse of that matrix does not exist! • Every square matrix has a determinant, however 2 x 2 matrices are the only ones we will calculate determinants for by hand • For larger matrices, finding the determinant is considerably more complicated (if you take a linear programming course in college, or AP Physics here at WHS, you may learn how to find 3 x 3 determinants by hand)

25. Finding the inverse of a 2 x 2 matrix • For the matrix: • The inverse is found by calculating: • In other words: • Switch the elements a and d • Reverse the signs of the elements b and c • Multiply by the fraction (1 / determinant)

26. Find the inverse of the matrix, if it exists. Example 7-2a Find the value of the determinant. Since the determinant is not equal to 0, S–1 exists.

27. Definition of inverse a = –1,b = 0,c = 8,d = –2 Answer: Simplify. Example 7-2b

28. Example 7-2c Check:

29. Find the inverse of the matrix, if it exists. Example 7-2d Find the value of the determinant. Answer: Since the determinant equals 0, T–1 does not exist.

30. Find the inverse of each matrix, if it exists. a.b. Answer: Example 7-2e Answer: No inverse exists.

31. Finding inverses for larger matrices • We will not calculate inverses of 3 x 3 or larger matrices by hand • However, we CAN find these using the TI-83 • Enter your matrix using the EDIT menu, then print it on your TI screen using the NAMES menu • Now hit the “X-1” button to indicate that you want to find the inverse of this matrix! • Let’s try some examples on the TI-83!!