Consider the matrix $A = \begin{bmatrix} 2&1&-1
5&2&-3
0&2&1 \end{bmatrix} $. Joe decides to send a message to Sam by encrypting the message with the matrix $A$. He first takes his message and converts it to numbers by replacing A with 1, B with 2, C with 3, and so on till replacing Z with 26. He uses a 0 for spaces. After replacing the letters with numbers, he breaks the message up into chunks of 3 letters. He then multiplies each chunk of 3 by the matrix $A$, resulting in a coded message. For example, to send the message “good job ben” he firsts converts the letters to the numbers and places them in a large matrix $M$ (top to bottom, left to right) $$ \left[ \begin{bmatrix}g\\o\\o\end{bmatrix}, \begin{bmatrix}d
\
j\end{bmatrix},\begin{bmatrix}o\\b\
\end{bmatrix},\begin{bmatrix}b\\e\\n\end{bmatrix}\right] \rightarrow \left[\begin{bmatrix}7\\15\\15\end{bmatrix}, \begin{bmatrix}4\\0\\10\end{bmatrix},\begin{bmatrix}15\\2\\0\end{bmatrix},\begin{bmatrix}2\\5\\14\end{bmatrix}\right] = M= \begin{bmatrix} 7 & 4 & 15 & 2
15 & 0 & 2 & 5
15 & 10 & 0 & 14 \end{bmatrix} .$$ To encode the matrix, he computes $$AM = \begin{bmatrix} 14 & -2 & 32 & -5
20 & -10 & 79 & -22
45 & 10 & 4 & 24 \end{bmatrix}.$$ and then sends the numbers $[ [ 14, 20, 45], [ -2, -10, 10], [ 32, 79, 4], [ -5, -22, 24]] $ to Sam. Sam uses the inverse of $A$ to decode the message.

1. Find the inverse of $A$.
2. Use $A^{-1}$ to decode $14, 20, 45],[ -2, -10, 10],[ 32, 79, 4],[ -5, -22, 24$ and show the message is ``good job ben. - Decode the message $39, 89, 22],[20, 48, 4],[39, 88, 33$. —- ==== Remarks ==== * Make remarks with a list. —- ==== $\LaTeX$ version ==== <file tex problem.simple_matrix_encryption.tex> % % DEPENDENCIES % RequiredPackages: \usepackage{tikz} % RequiredMacros: \DeclareMathOperator{\aut}{Aut} % %Encryption using matrices (and inverses to decode things). \begin{problem} Consider the matrix $A = \begin{bmatrix} 2&1&-1
5&2&-3
0&2&1 \end{bmatrix} $. Joe decides to send a message to Sam by encrypting the message with the matrix $A$. He first takes his message and converts it to numbers by replacing A with 1, B with 2, C with 3, and so on till replacing Z with 26. He uses a 0 for spaces. After replacing the letters with numbers, he breaks the message up into chunks of 3 letters. He then multiplies each chunk of 3 by the matrix $A$, resulting in a coded message. For example, to send the message ``good job ben he firsts converts the letters to the numbers and places them in a large matrix $M$ (top to bottom, left to right)

$$ \left[ \begin{bmatrix}g\\o\\o\end{bmatrix}, \begin{bmatrix}d
\
j\end{bmatrix},\begin{bmatrix}o\\b\
\end{bmatrix},\begin{bmatrix}b\\e\\n\end{bmatrix}\right] \rightarrow \left[\begin{bmatrix}7\\15\\15\end{bmatrix}, \begin{bmatrix}4\\0\\10\end{bmatrix},\begin{bmatrix}15\\2\\0\end{bmatrix},\begin{bmatrix}2\\5\\14\end{bmatrix}\right] = M= \begin{bmatrix} 7 & 4 & 15 & 2
15 & 0 & 2 & 5
15 & 10 & 0 & 14 \end{bmatrix} .$$ To encode the matrix, he computes $$AM = \begin{bmatrix} 14 & -2 & 32 & -5
20 & -10 & 79 & -22
45 & 10 & 4 & 24 \end{bmatrix}.$$ and then sends the numbers $[ [ 14, 20, 45], [ -2, -10, 10], [ 32, 79, 4], [ -5, -22, 24]] $ to Sam. Sam uses the inverse of $A$ to decode the message. \begin{enumerate} \item Find the inverse of $A$. \item Use $A^{-1}$ to decode $[ [ 14, 20, 45], [ -2, -10, 10], [ 32, 79, 4], [ -5, -22, 24]]$ and show the message is ``good job ben''. \item Decode the message $39, 89, 22],[20, 48, 4],[39, 88, 33$. \end{enumerate}

\end{problem}

</file>

problem