Solution

Encode the message "HILLCODE" with a Hill Cipher and encoding matrix given by

$$E=\left[ \begin{array}{cc} 1 & 13 \\ 2 & 7 \end{array} \right]$$

First, we translate "HILLCODE" into appropriate numbers $\pmod{26}$: $$\textrm{HILLCODE } \rightarrow 7,8,11,11,2,14,3,4$$

Then, treating each successive pair as a vector, we apply the given encoding matrix $\pmod{26}$:

$$\left[ \begin{array}{cc} 1 & 13 \\ 2 & 7 \end{array} \right] \left( \begin{array}{c} 7 \\ 8 \end{array} \right) = \left( \begin{array}{c} 111 \\ 70 \end{array} \right) \equiv \left( \begin{array}{c} 7 \\ 18 \end{array} \right)$$ $$\left[ \begin{array}{cc} 1 & 13 \\ 2 & 7 \end{array} \right] \left( \begin{array}{c} 11 \\ 11 \end{array} \right) = \left( \begin{array}{c} 154 \\ 99 \end{array} \right) \equiv \left( \begin{array}{c} 24 \\ 21 \end{array} \right)$$ $$\left[ \begin{array}{cc} 1 & 13 \\ 2 & 7 \end{array} \right] \left( \begin{array}{c} 2 \\ 14 \end{array} \right) = \left( \begin{array}{c} 184 \\ 102 \end{array} \right) \equiv \left( \begin{array}{c} 2 \\ 24 \end{array} \right)$$ $$\left[ \begin{array}{cc} 1 & 13 \\ 2 & 7 \end{array} \right] \left( \begin{array}{c} 3 \\ 4 \end{array} \right) = \left( \begin{array}{c} 55 \\ 34 \end{array} \right) \equiv \left( \begin{array}{c} 3 \\ 8 \end{array} \right)$$

Lastly, we re-interpret the resulting values as letters to arrive at the encrypted message: $$7,18,24,21,2,24,3,8 \rightarrow \textrm{ HSYVCYDI}$$