The question gives a $2\times2$ transition matrix:
$$
\begin{bmatrix}
0.8 & 0.2\\
0.3 & 0.7
\end{bmatrix}.
$$
And then it gives me the initial state matrix but I'm wondering how do I find the initial state matrix by myself?
The initial state matrix of this transition matrix is $$
\begin{bmatrix}
0.4 & 0.6\\
\end{bmatrix}.
$$
Best Answer
If you are given the final state matrix, then you could apply the inverse of the transition matrix to the final state matrix to get the initial state matrix. $$ \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}^{-1} =\begin{bmatrix} 1.4 & -0.4\\ -0.6 & 1.6 \end{bmatrix} $$
Normally, you are given the initial state matrix and you apply the transition matrix to get the final state matrix (which may become the initial state matrix for the next transition). For the given initial state matrix, the final state matrix would be $$ \begin{bmatrix} 0.4 & 0.6\\ \end{bmatrix} \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix} = \begin{bmatrix} 0.5 & 0.5\\ \end{bmatrix} $$