Answer
The matrix is $\left[ \begin{matrix}
33 & 93 \\
-21 & -61 \\
\end{matrix} \right]$ and the decoded message is $12,15,22,5$, or LOVE.
Work Step by Step
Consider the given expression, $\text{LOVE}$
$\begin{align}
& L=12 \\
& O=15 \\
& V=22 \\
& E=5 \\
\end{align}$
Using the cryptogram method, we get
$\begin{align}
& A=\left[ \begin{matrix}
4 & -1 \\
-3 & 1 \\
\end{matrix} \right]\left[ \begin{matrix}
12 & 22 \\
15 & 5 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
48-15 & 88+5 \\
-36+15 & -66+5 \\
\end{matrix} \right] \\
& =\left[ \begin{matrix}
33 & 93 \\
-21 & -61 \\
\end{matrix} \right]
\end{align}$
Therefore, expression of the matrix is $\left[ \begin{matrix}
33 & 93 \\
-21 & -61 \\
\end{matrix} \right]$
