Chapter 11.4, Problem 91AE

Cryptography One method of encryption is to use a matrix to encrypt the message and then use the corresponding inverse matrix to decode the message. The encrypted matrix, $E$, is obtained by multiplying the message matrix, $M$, by a key matrix, $K$. The original message can be retrieved by multiplying the encrypted matrix by the inverse of the key matrix.That is, $E=M\cdot K$ and $M=E\cdot K^{-1}$

(a) Given the key matrix $K=\left[\begin{array}{l}\begin{array}{ccc}2& 1& 1\end{array}\\ \begin{array}{ccc}1& 1& 0\end{array}\\ \begin{array}{ccc}1& 1& 1\end{array}\end{array}\right]$, find its inverse, $K^{-1}$. [Note: This key matrix is known as the $Q\begin{array}{c}3\\ 2\end{array}$ Fibonacci encryption matrix.]

(b) Use your result from part (a) to decode the encrypted matrix $E=\left[\begin{array}{l}\begin{array}{ccc}47& 34& 33\end{array}\\ \begin{array}{ccc}44& 36& 27\end{array}\\ \begin{array}{ccc}47& 41& 20\end{array}\end{array}\right]$

(c) Each entry in your result for part (b) represents the position of a letter in the English alphabet $K$ $\left(A=1,B=2,C=3,andsoon\right)$. What is the original message?

Source: goldenmuseum.com