1. 2.4 Inverse of Linear Transformations For an animation of this topic visit: http://www.maa.org/joma/Volume7/Hohenwarter/Transformations.html Is the transformation depicted In this picture invertible?

2. Ax=b Note our standard equation for the course is Ax=b However today we will look at the form xA=b In this case x must be a row vector to multiply by a matrix

3. Invertible Functions A function T from X to Y is called invertible if the equation T(x) = y has a unique solution x in X for each y in Y

4. Invertibility An nxn matrix is invertible if and only if a)rref of A is I b)Det(A) ≠ 0 c)There are no vectors other than the zero vector that satisfy the equation Ax=0 d)No row of A is a multiple of another row. No column of A is a multiple of another column No row of A is a linear combination of other rows of A No column of A is a linear combination of other columns of A e) rank (A) = n A non invertible matrix is called a singular matrix Note: Inverses are only defined for Square matrices

5. How to find an inverse In previous classes, we showed how to find the inverse of a matrix. We will not review that here. However, at there are 2 examples with step by step instructions of how to find the inverse of a matrix that you can review if you choose.

6. Cryptography An application of Inverses

7. Cryptography A Cryptogram is a message written according to a secret code. (The Greek word kryptos means hidden) If one wanted to write a secret message, one might first start by assigning a number to each letter of the alphabet (as shown on the next slide)

8. Encryption One might then break up a message into groups of letters for this example we will use blocks of 3 Next multiply each sequence by an encryption matrix

9. Continue this for each group of 3 terms

10. How would one decode this message?

11. One could use inverses to get the original terms back What would cause this system to not work properly? Please note that we are multiplying A -1 on the right side

12. Problem 26 Find the Inverse Note: you must follow a different process than the one taught previously, Why does our previous method fail to work?

13. 26 Solution

14. Recall:

15. How does the determinant of A relate to the determinant of A -1 Det(A) = 1/(detA -1 )

16. Homework P. 88 1-35 odd, 40, Pre-Calc book P. 608 29 - 39 odd,

17. Example 1 Find the inverse if it exists

18. Example 1 Solution Inverse does not exist

19. Example 2 Find the inverse if it exists

20. Example 2 Solution