Find the limit a matrix raised to $n$ when $n$ goes to infinity Let $ A $ be a $ 3\times3 $ matrix such that
$$A
\left( \begin{array}{ccc}
1 \\
2  \\
1  \end{array} \right)=\left( \begin{array}{ccc}
1 \\
2  \\
1  \end{array} \right),~~~A
\left( \begin{array}{ccc}
2 \\
2  \\
0  \end{array} \right)=\left( \begin{array}{ccc}
1 \\
1  \\
0  \end{array} \right),~~A
\left( \begin{array}{ccc}
3 \\
0  \\
6  \end{array} \right)=\left( \begin{array}{ccc}
-1 \\
0 \\
2  \end{array} \right)
$$
Find $$ \lim_{n\to\infty}A^n \left( \begin{array}{ccc}
6 \\
7  \\
0  \end{array} \right)$$
So, do I first find $ A $ by letting A =$ \left( \begin{array}{ccc}
a&b&c \\
d&e&f  \\
g&h&i  \end{array} \right) $ and using the given information to solve the corresponding linear equations and then solve the actual problem of finding the limit? Is there a more efficient way of doing this? Also, I am not quite sure how to find the limit so any hints would be greatly appreciated. Thanks!

Use diagonalization. $A=PDP^{-1}$, where $P=\pmatrix{1&2&3\cr 2&2&0\cr 1&0&6\cr}$ and $D=\pmatrix{1&0&0\cr 0&1/2&0\cr 0&0&-1/3\cr}$. Then
$$\lim_{n\to\infty} A^n \pmatrix{6\cr 7\cr 0\cr} = \lim_{n\to\infty} (PD^n P^{-1}) \cdot \pmatrix{6\cr 7\cr 0\cr} = P \lim_{n\to\infty} D^n \cdot P^{-1} \cdot \pmatrix{6\cr 7\cr 0\cr} = P \cdot \pmatrix{1&0&0\cr 0&0&0\cr 0&0&0\cr} \cdot P^{-1} \cdot \pmatrix{6\cr 7\cr 0\cr}.$$
