As explained in <https://en.wikipedia.org/wiki/Markov_chain>, for a three state Markov Chain with the transition matrix given as

$P = \begin{bmatrix}
0.9 & 0.075 & 0.025 \\
0.15 & 0.8 & 0.05 \\
0.25 & 0.25 & 0.5
\end{bmatrix},$

the steady state probabilities can be found from

$\lim\_{N\to \infty } \, P^N=
\begin{bmatrix}
 0.625 & 0.3125 & 0.0625 \\
 0.625 & 0.3125 & 0.0625 \\
 0.625 & 0.3125 & 0.0625 \\
\end{bmatrix}.$

Now my Question is, how can I find a valid transition matrix P for any other steady state probabilities, lets say for example $[0.3, 0.5, 0.2]?$.
Preferably with P containing only elements that are nice to work with by hand.