The transition matrix for a Markov chain is shown to the right.
\[
P=\begin{array}{l}
A \\
B
\end{array}\left[\begin{array}{ll}
0.4 & 0.6 \\
0.9 & 0.1
\end{array}\right]
\]
Find $P^{k}$ for $k=2,4$, and 8 . Can you identify a matrix $Q$ that the matrices $P^{k}$ are approaching?

Step 1 ：The given transition matrix for a Markov chain is \(P = \begin{bmatrix} 0.4 & 0.6 \\ 0.9 & 0.1 \end{bmatrix}\).

Step 2 ：We are asked to find the powers of this matrix for \(k = 2, 4, 8\).

Step 3 ：By multiplying the matrix by itself for the given powers, we get \(P^2 = \begin{bmatrix} 0.7 & 0.3 \\ 0.45 & 0.55 \end{bmatrix}\), \(P^4 = \begin{bmatrix} 0.625 & 0.375 \\ 0.5625 & 0.4375 \end{bmatrix}\), and \(P^8 = \begin{bmatrix} 0.6015625 & 0.3984375 \\ 0.59765625 & 0.40234375 \end{bmatrix}\).

Step 4 ：Observing these matrices, it appears that as the power increases, the values in the matrix are converging towards a certain value. This suggests that there is a matrix \(Q\) that the matrices \(P^k\) are approaching.

Step 5 ：We can approximate this matrix \(Q\) by taking the average of the values in the same positions across the matrices \(P^2\), \(P^4\), and \(P^8\).

Step 6 ：Doing so, we find that \(Q = \begin{bmatrix} 0.6421875 & 0.3578125 \\ 0.53671875 & 0.46328125 \end{bmatrix}\).

Step 7 ：\(\boxed{Q = \begin{bmatrix} 0.6421875 & 0.3578125 \\ 0.53671875 & 0.46328125 \end{bmatrix}}\) is the matrix that the matrices \(P^k\) are approaching.