Mathematics - Matrices Question with Solution | TestHub

$$\begin{gathered} P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right] \\ \Rightarrow P^2=\left[\begin{array}{ccc}1& 0& 0\\ 8& 1& 0\\ 16+32& 8& 1\end{array}\right] \end{gathered}$$
So,$P^3=\left[\begin{array}{ccc}1& 0& 0\\ 12& 1& 0\\ 16+32+48& 12& 1\end{array}\right]$
Using symmetry, we can write
$P^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 200& 1& 0\\ \frac{\mathrm{16.50.51}}{2}& 200& 1\end{array}\right]$
As,$P^{50}-Q=I\Rightarrow Q=P^{50}-I=\left[\begin{array}{ccc}0& 0& 0\\ 200& 0& 0\\ \frac{\mathrm{16.50.51}}{2}& 200& 0\end{array}\right],$
$q_{31}=\frac{\mathrm{16.50.51}}{2},$$q_{32}=200$and$q_{21}=200$
$\therefore$$\frac{q_{31}+q_{32}}{q_{21}}=\frac{\mathrm{16.50.51}}{2.200}+1=102+1=103$