For a Markov chain with two states and the transition probability matrix $\mathbf{P} = \left[\begin{array}{cc} 0.1 &0.9\cr 0.5 &0.5 \end{array}\right]$ Find the matrix of the mean first-passage times $\mu_{ij}=\mathbb{E}[T_{ij}]$
So far I have tried using the formula $m_{ij} = 1 + \sum_{k \neq j} p_{ik}m_{kj}$ for $i \neq j$ of the matrix and the formula $m_{ii} = \frac{1}{\pi_i}$ for $i = j$ entries of the matrix where $\pi_i$ is an entry in the steady state vector which I have calculated to be $\pi = \begin{bmatrix}\frac{0.45}{1.45}&\frac{1}{1.45}\end{bmatrix}$ and came up with the matrix $\begin{bmatrix}\frac{1.45}{0.45}&\frac{1}{0.9} \\\ 2&1.45 \end{bmatrix}$ which was marked incorrect
I would appreciate any help leading me in the right direction.
the stationary distribution of matrix $P$ is $\pi = {\frac{5}{14},\frac{9}{14}}$ .The mean first passage matrix is then : $\mathbf{M_{\pi}} = \left[\begin{array}{cc} 0 &\frac{1}{0.9}\cr 2 &0 \end{array}\right]$