Let P=pmatrix1&0&03&1&09&3&1pmatrix and Q=[q_ij] be a 33 matrix such that Q-P^5=I_3.

Question

Let $P=\begin{pmatrix}1&0&0\3&1&0\9&3&1\end{pmatrix}$ and $Q=[q_{ij}]$ be a $3	imes3$ matrix such that $Q-P^5=I_3$. Then $\dfrac{q_{21}+q_{31}}{q_{32}}$ is equal to:

Accepted Answer

By repeated squaring, $P^2=\begin{pmatrix}1&0&0\6&1&0\27&6&1\end{pmatrix}$, $P^4=\begin{pmatrix}1&0&0\12&1&0\90&12&1\end{pmatrix}$, and $P^5=P^4P=\begin{pmatrix}1&0&0\15&1&0\135&15&1\end{pmatrix}$.

$Q=I_3+P^5=\begin{pmatrix}2&0&0\15&2&0\135&15&2\end{pmatrix}$, so $q_{21}=15,\ q_{31}=135,\ q_{32}=15$.

$$\frac{q_{21}+q_{31}}{q_{32}}=\frac{15+135}{15}=10.$$

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