<s>[INST] limit law and product of matrices Are there two $ n\times n $ matrices $A$ and $B$ such that $ \lim_{m\to\infty} A^m$  and $ \lim_{m\to\infty} B^m  $ both exists but $ \lim_{m\to\infty} (A \cdot B)^m  $ doesn't ?  
 [/INST] Consider 
$$
   A = \begin{pmatrix}  0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0\end{pmatrix}   \qquad B = A^T 
$$
Then $\forall m \geqslant 3$, $A^m = B^m = 0$, but since
$$
   A \cdot B = \begin{pmatrix}  0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 2\end{pmatrix}
$$
$$
\lim_{m \to \infty} (A \cdot B)^m  = \lim_{m \to \infty} \begin{pmatrix} 
 0 & 0 & 0 \\
 0 & \phi ^{2-2 m}+\phi ^{2 m-2} & -\phi ^{1-2 m}+\phi ^{2 m-1} \\
 0 & -\phi ^{1-2 m}+\phi ^{2 m-1} & \phi ^{-2 m}+\phi ^{2 m} \end{pmatrix} = 
\begin{pmatrix}  0 & 0 & 0 \\ 0 & \infty & \infty \\ 0 & \infty & \infty\end{pmatrix}
$$
</s>