> 
> Question: Let $I$ be an identity matrix of order $2 \times 2$ and $P = \begin{bmatrix}
>  2 & -1 \\
>  5 & -3 \\
>  \end{bmatrix}$. Then the value of $n ∈ N$ for which $P^n = 5I - 8P$ is equal to \_\_\_\_\_\_\_.
> 
> 
> Answer: 6
> 
> 
> Question Source: JEE Mains $18^{th}$ March Shift-2 2021
> 
> 
> 

By characteristic Equation: $P^2-(\operatorname {Tr}(P))P+(\det (P))I=0$ we can find $n=6$ by hit and trial. i.e. multiplying equation by P gives

$P^3=(\operatorname {Tr}(P))P^2-(\det (P))P = (\operatorname {Tr}(P))[(\operatorname {Tr}(P))P+(\det(P))I]$

And going on solving for $P^6$. But is there a generalized method because it can't be done for high values of $n$ (eg:$100$)?