Let A be the $3×3$ matrix $$ A=\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}. $$
Determine all real numbers $a$ for which the limit $$\lim_{n\to \infty}a^nA^n$$ exists and is non zero.
My answer. I think $a=1$, as I was not able to think how to tackle this kind of problem. Please help me or give me some hints or a solution.
I would be very thankful.
The matrix $A$ is diagonalizable and its eigenvalues are $0$, $1$, and $3$. Therefore, the answer is $a=\frac13$. In fact, since there is an invertible $3\times3$ matrix $P$ such that$$A=P\begin{pmatrix}3&0&0\\0&1&0\\0&0&0\end{pmatrix}P^{-1},$$then, for each $a\in\mathbb R$ and each $n\in\mathbb N$,$$a^nA^n=P\begin{pmatrix}(3a)^n&0&0\\0&a^n&0\\0&0&0\end{pmatrix}P^{-1}$$and the sequence$$\begin{pmatrix}(3a)^n&0&0\\0&a^n&0\\0&0&0\end{pmatrix}_{n\in\mathbb N}$$has a non-zero limit if and only if $a=\frac13$.