Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 8& 4& 1\end{array}\right]$ and I be the identity matrix of order 3. If $\mathrm{Q}=\left[{\mathrm{q}}_{\mathrm{ij}}\right]$ is a matrix such that  ${\mathrm{p}}^{500}-\mathrm{Q}=\mathrm{I}\text{, then}\frac{{\mathrm{q}}_{31}+{\mathrm{q}}_{32}}{{\mathrm{q}}_{21}}\text{ equals}$

$\mathrm{P}=\mathrm{I}+\mathrm{B},\mathrm{B}=\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]$

${\mathrm{B}}^2=\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 16& 0& 0\end{array}\right]$

${\mathrm{B}}^3=0$

${\mathrm{P}}^{500}=(\mathrm{I}+\mathrm{B}{)}^{500}=\mathrm{I}+500\mathrm{B}+250\times 499{\mathrm{B}}^2$

$\Rightarrow {\mathrm{P}}^{500} \mathrm{I}-I=500\mathrm{B}+250\times 499{\mathrm{B}}^2$

$\frac{500\times 8+250\times 499\times 16+500\times 4}{500\times 4+0}=1001$