For the transition matrix P=
0.3 0.7 00
0.5 0.1 0.4
, solve the equation SP = S to find the stationary matrix S and the limiting matrix P.
0.0 0.3 0.7

Step 1 ：Given the transition matrix P = \(\begin{bmatrix} 0.3 & 0.7 & 0 \ 0.5 & 0.1 & 0.4 \ 0 & 0.3 & 0.7 \end{bmatrix}\)

Step 2 ：The stationary matrix S is a row vector such that S*P = S, where P is the transition matrix. This means that the stationary matrix is the eigenvector of the transition matrix corresponding to the eigenvalue 1.

Step 3 ：To find the stationary matrix, we need to solve the equation (S - I)P = 0, where I is the identity matrix. This is equivalent to finding the null space of the matrix (S - I)P.

Step 4 ：The limiting matrix P is the matrix that the transition matrix approaches as the number of transitions approaches infinity. This is equivalent to raising the transition matrix to a large power.

Step 5 ：By solving the above equations, we find that the stationary matrix S is \(\boxed{\begin{bmatrix} 0.33333333 \\ 0.33333333 \\ 0.33333333 \end{bmatrix}}\)

Step 6 ：And the limiting matrix P is \(\boxed{\begin{bmatrix} 0.234375 & 0.328125 & 0.4375 \\ 0.234375 & 0.328125 & 0.4375 \\ 0.234375 & 0.328125 & 0.4375 \end{bmatrix}}\)