Question: Secret messages can be encoded by using a code and an encoding matrix. Suppose we have the following code:

We encode an empty space by 0. Let the encoding matrix be

\[E=\left[ \begin{matrix} 1 & -2 \\ -2 & 3 \\ \end{matrix} \right]\]

Then we can encode a message by taking every two letters of the message, converting them to their corresponding numbers, creating a 2 × 1 matrix, and then multiplying each matrix by E.

(a) Use this matrix to encode the message “Happy new year”

(b) Use this code and matrix to decode

\[\left[ \begin{matrix} 13 \\ -31 \\ \end{matrix} \right],\left[ \begin{matrix} 6 \\ -15 \\ \end{matrix} \right],\left[ \begin{matrix} -11 \\ 9 \\ \end{matrix} \right],\left[ \begin{matrix} 5 \\ -10 \\ \end{matrix} \right],\left[ \begin{matrix} 0 \\ -1 \\ \end{matrix} \right],\left[ \begin{matrix} -19 \\ 27 \\ \end{matrix} \right]\]