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!
Best Answer
Hint: Write the vector $(6,7,0)^{T}$ as a linear combination of $(1,2,1)^{T}$, $(2,2,0)^{T}$, and $(3,0,6)^{T}$. Then use the formulas from the first line.
This will give an equation like $A(c_{1}v_{1}+c_{2}v_{2}+c_{3}v_{3}) = c_{1}Av_{1}+c_{2}Av_{2}+c_{3}Av_{3}$. Your formulas at the beginning will give a nice way to write $Av_{i}$ (for example, $Av_{1}=v_{1}$), and should give a clue on how to get an answer for general $n$. You can take the limit from here.