1. The inverse of a Square Matrix 2.1 Day 1 (Out of Pre-Calc book 8.3) We are reloading for another chapter

2. Definition of Inverse The inverse of a matrix is the matrix that you multiply times it to obtain I A*A -1 = I = A -1 A Note: matrix multiplication is NOT generally commutative but it is in the case of square invertible matrices. Our books defines inverses for square matrices some books include inverses for rectangular matrices (would these be commutative?)

3. Example 1 (pre-Calc)

4. Example 1 solution

5. To solve Ax=b One method would be to multiply both sides of the equation by A inverse. (the matrix that when multiplied times A yields the identity matrix.) 1 3 2 6

6. How to find an inverse

7. Inverse in terms of Determinants (p. 586) For a 2x2 matrix we can find the inverse of a matrix using determinants

8. Example 3 Find the inverse of the matrix

9. Example 3 solution

10. What causes an inverse to fail to exist? One column is a multiple of another column If the determinant = 0 If there are any non zero vectors x that are solutions to the equation Ax= 0 If any row is a multiple of another row

11. Example 5 Solve the system of equations using the inverse of a matrix. Use technology to check your answer.

12. Example 5 Solution First, write the system in the form Ax=b Next, find A inverse Finally, multiple both sides of the equation by A inverse When would it be beneficial to use this method to solve systems of equations?

13. Example 5 Solve the system of equations using the inverse of a matrix. Use technology to check your answer. To check the final answer use rref as shown in a previous lecture. To find the inverse of a matrix on a TI 89 Calculator. Enter the matrix and raise it to the -1 power

14. Homework p. 588 (pre-Calc) 1-63 every other odd 1 0 0 0 1 0 0 0 1 [ ] Superman = Clark Kent By: Mr. Whitehead