Answer
The decoded message is $18,\ 21\,12,\ 5$ or a RULE
Work Step by Step
First encode the given message as below:
The word RULE is numerically equivalent to $18,21,12\text{ and 5}$.
Now write these entries in the form of a square matrix as below:
$\left[ \begin{matrix}
18 & 21 \\
12 & 5 \\
\end{matrix} \right]$
Multiplying this matrix by the square matrix $ A=\left[ \begin{matrix}
3 & 2 \\
4 & 3 \\
\end{matrix} \right]$ we get:
$\begin{align}
& \left[ \begin{matrix}
3 & 2 \\
4 & 3 \\
\end{matrix} \right]\left[ \begin{matrix}
18 & 21 \\
12 & 5 \\
\end{matrix} \right]=\left[ \begin{matrix}
3\left( 18 \right)+2\left( 21 \right) & 3\left( 12 \right)+2\left( 5 \right) \\
4\left( 18 \right)+3\left( 21 \right) & 4\left( 12 \right)+3\left( 5 \right) \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
96 & 46 \\
135 & 63 \\
\end{matrix} \right]
\end{align}$
Now use these numbers, by columns, to write the encoded message $96,135,46,63$.
Next decode this message by multiplying the multiplicative inverse of the coding matrix and the coded matrix as given below:
It is given that the inverse of the coding matrix is, ${{A}^{-1}}=\left[ \begin{matrix}
3 & -2 \\
-4 & 3 \\
\end{matrix} \right]$
