1. 4.5 Matrices, Determinants, Inverseres -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. Example 7-1a Determine whether X and Y are inverses.

8. Example 7-1b

9. Example 7-1c Determine whether P and Q are inverses.

10. Example 7-1d Determine whether each pair of matrices are inverses. a. b.

11. How do we find the inverse??? 1 st find the determinant The determinant; – determines whether the inverse of a matrix exists. –influences the elements the inverse contains For the matrix shown below, the determinant is equal to ad – bc In words, multiply the elements in each diagonal, then subtract the products! Order Matters.

15. 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. We will use DETERMINANTS and INVERSES to solve matrix equations of the type AX = B.

16. 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)

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

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

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

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

22. Example 7-2c Check: