> 
> Given is markov chain $\left\{X\_n\right\}\_{n \in \mathbb{N}}$ with
>  transition probabilities
> 
> 
> $$M= \begin{pmatrix} 
> 1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 
> 1/4 & 3/4 & 0 & 0 & 0 & 0 \\ 
> 1/4 & 1/4 & 1/4 & 1/4 & 0 & 0 \\ 
> 1/4 & 0 & 1/4 & 1/4 & 0 & 1/4 \\ 
> 0 & 0 & 0 & 0 & 1/2 & 1/2 \\ 
> 0 & 0 & 0 & 0 & 1/2 & 1/2 
> \end{pmatrix}$$
> 
> 
> Determine the probability $f\_1(n)$ where you return to state $1$ after
>  $n$ steps (for the first time).
> 
> 
> 

I'm not sure how you can solve this because if I understood it correctly, we have $n$ steps and we are looking for a probability, so we have two unknowns...

Anyway, I think the correct way of calculating it is (don't miss the little exponent $n$ of that huge matrix!)

$$f\_1(n) = \begin{pmatrix} 
1/2 & 1/2 & 0 & 0 & 0 & 0 \\ 
1/4 & 3/4 & 0 & 0 & 0 & 0 \\ 
1/4 & 1/4 & 1/4 & 1/4 & 0 & 0 \\ 
1/4 & 0 & 1/4 & 1/4 & 0 & 1/4 \\ 
0 & 0 & 0 & 0 & 1/2 & 1/2 \\ 
0 & 0 & 0 & 0 & 1/2 & 1/2 
\end{pmatrix}^n \cdot 
\begin{pmatrix}
1\\ 
0\\ 
0\\ 
0\\ 
0\\
0
\end{pmatrix}$$

Here I'm stuck again for the same reason, I see no way of calculating the probability...? : /