<s>[INST] Unclear problem with $n$-th power matrix and limit 
Find $$\lim\limits_{n \to \infty} \frac{A_n}{D_n}$$ where
$$\begin{pmatrix}    
    19 & -48 \\
    8 & -21 \\
    \end{pmatrix} ^{\! n} =  
    \begin{pmatrix}
    A_n & B_n \\
    C_n & D_n \\
    \end{pmatrix}$$


$n$ - is the power of a matrix, but what is $A_n, B_n, C_n, D_n$ then? Is it a corresponding element of a matrix in the $n$-th power?
How is this type of problem called? And what is the way to solve that problem?
 [/INST] $$A:=\begin{pmatrix}19&-48\\8&-21\end{pmatrix}=\begin{pmatrix}2&3\\1&1\end{pmatrix}\begin{pmatrix}-5&0\\0&3\end{pmatrix}\begin{pmatrix}2&3\\1&1\end{pmatrix}^{-1}=PDP^{-1}$$
So $A^n=PDP^{-1}PDP^{-1}\cdots PDP^{-1}=PD^nP^{-1}$
That is, $$A^n=\begin{pmatrix}2&3\\1&1\end{pmatrix}\begin{pmatrix}(-5)^n&0\\0&3^n\end{pmatrix}\begin{pmatrix}2&3\\1&1\end{pmatrix}^{-1}=\begin{pmatrix}-2 (-5)^n + 3^{1 + n}& 6 (-5)^n - 2\times 3^{1 + n}\\-(-5)^n + 3^n& 
 3 (-5)^n - 2\times 3^n\end{pmatrix}$$
</s>