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?

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

Invertible FunctionsA 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

InvertibilityAn nxn matrix is invertible if and only ifa) rref of A is Ib) Det(A) ≠ 0c) There are no vectors other than the zero vector that

satisfy the equation Ax=0d) 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

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.

Cryptography

An application of Inverses

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)

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

Continue this for each group of 3 terms

How would one decode this message?

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

Problem 26

Find the InverseNote: you must follow a different process

than the one taught previously,Why does our previous method fail to work?

26 Solution

Recall:

How does the determinant of A relate to the determinant of A-1

Det(A) = 1/(detA-1)

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

Example 1Find the inverse if it exists

Example 1 Solution

Inverse does not exist

Example 2Find the inverse if it exists

Example 2 Solution