I have a matrix in the form :

$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A 
\end{pmatrix}
$$

where $A,B,C,D$ are diagonalizable square matrices and I want to determine

$$M^{\infty}:=\lim_{n\rightarrow \infty} M^n$$ 

as a function of $A,B,C,D$.

Thank you for your help !