Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals
Step-by-Step Solution
Key Concept: Compute successive powers of $P$ to find patterns, then use matrix inverse properties to determine the required entries.
Given $P = \begin{bmatrix}1 & 0 & 0\\4 & 1 & 0\\16 & 4 & 1\end{bmatrix}$, we compute powers: $P^2, P^3, \ldots, P^{50}$ showing $P^{50} - Q = I$ where $Q = P^{50} - I$. The inverse satisfies $P^{-1}Q = \begin{bmatrix}1 & 0 & 0\\200 & 1 & 0\\20400 & 200 & 1\end{bmatrix}P^{-1}$. Solving for entries: $q_{21} = 200$, $q_{31} = 20400$, $q_{32} = 200$, giving $\frac{q_{31}+q_{32}}{q_{21}} = \frac{20600}{200} = 103$.
Correct Answer: 103