$P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right] $

$ P^2=\left[\begin{array}{ccc}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}{ccc}1 & 0 & 0 \\ 8 & 1 & 0 \\ 48 & 8 & 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 \\ 12 & 1 & 0 \\ 56 & 12 & 1\end{array}\right]$

$P^4=\left[\begin{array}{ccc}1 & 0 & 0 \\ 8 & 1 & 0 \\ 48 & 8 & 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 \\ 16 & 1 & 0 \\ 160 & 16 & 1\end{array}\right] $

$\therefore P ^{50}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 \times 50 & 1 & 0 \\ p _{31} & 4 \times 50 & 1\end{array}\right] Q =\left[\begin{array}{lll} q _{11} & q _{12} & q _{13} \\ q _{21} & q _{22} & q _{23} \\ q _{31} & q _{32} & q _{33}\end{array}\right]$

$ P^{50}-Q=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

$\Rightarrow 4 \times 50- q _{21}= o$

$p _{31}- q _{31}= o$

$4 \times 50- q _{32}= q$

On observing,

$p _{31^1}=16 \times 1$

$p _{31^2}=16 \times(1+2)$

$p _{31^3}=16 \times(1+2+3)$

$p _{3150}=16 \times(1+2+3+\ldots .50)$

$=16 \times \frac{50 \times 51}{2}$

$\therefore \frac{ q _{31}+ q _{32}}{ q _{21}}=\frac{ p _{31}+ p _{32}}{ p _{21}}$

$=\frac{16 \times 25 \times 51+4 \times 50}{4 \times 50}$

$=103$