Approximate the stationary matrix $S$ for the transition matrix $P$ by computing powers of the transition matrix $P$. \[ P=\left[\begin{array}{rr} 0.46 & 0.54 \\ 0.2 & 0.8 \end{array}\right] \]

Step 1 ：Given the transition matrix \(P = \left[\begin{array}{rr} 0.46 & 0.54 \\ 0.2 & 0.8 \end{array}\right]\)

Step 2 ：We need to find the stationary matrix \(S\) for this transition matrix. The stationary matrix \(S\) is a matrix such that \(SP = S\). In other words, the stationary matrix remains unchanged when the transition matrix is applied to it.

Step 3 ：One way to approximate the stationary matrix is by computing powers of the transition matrix until the results converge. This is based on the Perron-Frobenius theorem, which states that for a stochastic matrix (like our transition matrix), the power of the matrix will converge to a matrix where each row is the stationary distribution.

Step 4 ：We can compute powers of the transition matrix until the results converge. We will stop the computation when the difference between two consecutive powers of the matrix is less than a small threshold (e.g., 0.0001).

Step 5 ：After performing the computation, we find that the power of the transition matrix converges to \(P_{\text{power}} = \left[\begin{array}{rr} 0.27032888 & 0.72967112 \\ 0.27024856 & 0.72975144 \end{array}\right]\)

Step 6 ：From this, we can see that the stationary matrix \(S\) is approximately \(S = [0.27032888, 0.72967112]\)

Step 7 ：Final Answer: The stationary matrix \(S\) for the transition matrix \(P\) is approximately \(\boxed{S = [0.27032888, 0.72967112]}\)