Answer
$\lim _{k \rightarrow \infty} A^{k}=\frac{1}{3}\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right]$
Work Step by Step
Using MATLAB, find $A^{k}$, for $k=5,10,20,30$ :
\[
\begin{aligned}
A^{5}=\left[\begin{array}{ccc}
0.3318 & 0.3346 & 0.3336 \\
0.3346 & 0.3323 & 0.3331 \\
0.3336 & 0.3331 & 0.3333
\end{array}\right] \\
A^{10}=A^{20}=A^{30}=\left[\begin{array}{lll}
0.3333 & 0.3333 & 0.3333 \\
0.3333 & 0.3333 & 0.3333 \\
0.3333 & 0.3333 & 0.3333
\end{array}\right]
\end{aligned}
\]
So,as power becomes bigger and bigger, the matrix becomes more and more like
\[
\frac{1}{3}\left[\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
\]
