Question

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 $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I,$ then the value of $\frac{q_{31}+q_{32}}{q_{21}}$ is equal to

• A. $52$
• B. $103$
• C. $201$
• D. $205$

Answer 1

Answer: (B)

$P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]\Rightarrow P^2=\left[\begin{array}{ccc}1& 0& 0\\ 8& 1& 0\\ 16+32& 8& 1\end{array}\right]$
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 $q_{21}=200$
$\therefore$ $\frac{q_{31}+q_{32}}{q_{21}}=\frac{\mathrm{16.50.51}}{2.200}+1=102+1=103$