Consider the Markov chain with transition matrix
\[ P = \begin{pmatrix} 1/2 & 1/2 \\ 1/4 & 3/4 \end{pmatrix} \]
Find the fundamental matrix Z for this chain. Compute the mean first passage matrix using Z.
\[ (P - I)Z = 0 \]
\[ P - I \]:
\[ P - I = \begin{pmatrix} 1/2 - 1 & 1/2 \\ 1/4 & 3/4 - 1 \end{pmatrix} = \begin{pmatrix} -1/2 & 1/2 \\ 1/4 & -1/4 \end{pmatrix} \]
Fundamental matrix = \((P - I)^{-1}\):
\[ (P - I)^{-1} = \begin{pmatrix} -1/2 & 1/2 \\ 1/4 & -1/4 \end{pmatrix}^{-1} = \frac{1}{(-1/2)(-1/4) - (1/2)(1/4)} \begin{pmatrix} -1/4 & -1/2 \\ -1/4 & -1/2 \end{pmatrix} \]
\[ = \frac{4}{1} \begin{pmatrix} -1/4 & -1/2 \\ -1/4 & -1/2 \end{pmatrix} = \begin{pmatrix} -1 & -2 \\ -1 & -2 \end{pmatrix} \]