Markov Chains & Long-Run Behavior · Hard problem

A chain has transition matrix $M = \begin{pmatrix} 0.6 & 0.3 \\ 0.4 & 0.7 \end{pmatrix}$, where $M_{ij}$ is the probability of moving to state $i$ from state $j$. Compare starting in state $1$ to starting in state $2$: after $2$ steps, which start gives the higher probability of being in state $1$, and what is the positive difference between the two probabilities? Express your answer as a common fraction.