$A+{{A}^{2}}+{{A}^{3}}+I=\left[ \begin{matrix}
0 & 0 & 0 \\
\alpha & 0 & 0 \\
\beta & \gamma & 0 \\
\end{matrix} \right]+\left[ \begin{matrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
\alpha \gamma & 0 & 0 \\
\end{matrix} \right]+\bar{O}+\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right]=\left[ \begin{matrix}
1 & 0 & 0 \\
\alpha & 1 & 0 \\
\alpha \gamma +\beta & \gamma & 1 \\
\end{matrix} \right]$
پس:
$\left| A+{{A}^{2}}+{{A}^{3}}+I \right|=\left[ \begin{matrix}
1 & 0 & 0 \\
\alpha & 1 & 0 \\
\alpha \gamma +\beta & \gamma & 1 \\
\end{matrix} \right]=1$