Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and I be the identity matrix of order 3. If Q = [q_ij] is a matrix such that $P^{50}-Q=I,\text{ then }\frac{q_{31}+q_{32}}{q_{21}}\text{=}$

$\begin{array}{l}{P}^2=\left[\begin{array}{lll}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 8& 1& 0\\ 48& 8& 1\end{array}\right]\\ P^3=\left[\begin{array}{lll}1& 0& 0\\ 8& 1& 0\\ 48& 8& 1\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 12& 1& 0\\ 96& 12& 1\end{array}\right]\\ \therefore P^n=\left[\begin{array}{ccc}1& 0& 0\\ 4n& 1& 0\\ 8\left(n^2+n\right)& 4n& 1\end{array}\right]\\ \therefore P^{50}=\left[\begin{array}{ccc}200& 1& 0\\ 8\times 50\left(52\right)& 200& 1\end{array}\right]\Rightarrow p^{50}-Q=I\\ \therefore \text{ Equating we get }{q}_{21}=200\\ \Rightarrow q_{31}=400\times 51\Rightarrow q_{32}=200\\ \Rightarrow \frac{q_{31}+q_{32}}{q_{21}}=\frac{400\times 51+200}{200}=2\left(51\right)+1=103\end{array}$