Please note down all invariant distributions for each of the following transition matrices.
$\begin{pmatrix} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{2}&\frac{1}{2}&0 \end{pmatrix}$, $\begin{pmatrix} 1&0&0&0&\dots\\ 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ 0&0&0&1&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$, $\begin{pmatrix} 1&0&0&0&\dots\\ 1&0&0&0&\dots\\ 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$, $\begin{pmatrix} 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ 0&0&0&1&\dots\\ 0&0&0&0&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$
I tried to set up a system of equations for the first matrix. Is the invariant distribution $\begin{pmatrix} \frac{2}{7}\\ \frac{3}{7}\\ \frac{2}{7} \end{pmatrix}$ for the first matrix correct? Are there more possibilities?
I also did it for the second, third and fourth matrix. For each of these three matrices I got the invariant distribution $\begin{pmatrix} \frac{1}{t}\\ \frac{1}{t}\\ \frac{1}{t}\\ \frac{1}{t}\\ \vdots \end{pmatrix}$. Is that correct? Are there more possibilities?