$\mathrm{A}=\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\end{array}\right]$ and $(\left.{\mathrm{A}}^8+{\mathrm{A}}^6+{\mathrm{A}}^4+{\mathrm{A}}^2+\mathrm{I}\right)\mathrm{V}=\left[\begin{array}{c}0\\ 11\end{array}\right]$ (where I is the 2 x 2 identity matrix), then the product of all elements of matrix V is ________.

$\begin{array}{l}\mathrm{A}=\left[\begin{array}{ll}0& 1\\ 3& 0\end{array}\right]\\ \Rightarrow {\mathrm{A}}^2=\mathrm{A}\cdot \mathrm{A}=\left[\begin{array}{ll}0& 1\\ 3& 0\end{array}\right]\left[\begin{array}{ll}0& 1\\ 3& 0\end{array}\right]=\left[\begin{array}{ll}3& 0\\ 0& 3\end{array}\right]\\ \Rightarrow {\mathrm{A}}^4={\mathrm{A}}^2\cdot {\mathrm{A}}^2=\left[\begin{array}{ll}3& 0\\ 0& 3\end{array}\right]\left[\begin{array}{ll}3& 0\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}{3}^2& 0\\ 0& 3^2\end{array}\right]\end{array}$

$\Rightarrow {\mathrm{A}}^8=\left[\begin{array}{cc}{3}^4& 0\\ 0& 3^4\end{array}\right]$

and ${\mathrm{A}}^6={\mathrm{A}}^4\cdot {\mathrm{A}}^2=\left[\begin{array}{cc}{3}^2& 0\\ 0& 3^2\end{array}\right]\left[\begin{array}{l}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right]=\left[\begin{array}{cc}{3}^3& 0\\ 0& 3^3\end{array}\right]$

Let $\mathrm{V}=\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]$

$\begin{array}{l}{\mathrm{A}}^8+{\mathrm{A}}^6+{\mathrm{A}}^4+{\mathrm{A}}^2+\mathrm{I}\\ \left[\begin{array}{cc}81& 0\\ 0& 81\end{array}\right]+\left[\begin{array}{cc}27& 0\\ 0& 27\end{array}\right]+\left[\begin{array}{ll}9& 0\\ 0& 9\end{array}\right]+\left[\begin{array}{ll}3& 0\\ 0& 3\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ =\left[\begin{array}{cc}121& 0\\ 0& 121\end{array}\right]\\ \left({\mathrm{A}}^8+{\mathrm{A}}^6+{\mathrm{A}}^4+{\mathrm{A}}^2+\mathrm{I}\right)\mathrm{V}=\left[\begin{array}{c}0\\ 11\end{array}\right]\\ \mathrm{or} \left[\begin{array}{cc}121& 0\\ 0& 121\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}0\\ 11\end{array}\right]\\ \mathrm{or} \left[\begin{array}{c}121\mathrm{x}\\ 121\mathrm{y}\end{array}\right]=\left[\begin{array}{c}0\\ 11\end{array}\right]\\ \mathrm{x}=0\text{ and }\mathrm{y}=1/11\\ \mathrm{V}=\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}0\\ 1/11\end{array}\right]\end{array}$