Markov Chains & Long-Run Behavior · Medium problem

A chain has transition matrix $M = \begin{pmatrix} 1/2 & 1/4 \\ 1/2 & 3/4 \end{pmatrix}$ ($M_{ij}$ is the probability of moving to state $i$ from state $j$) and starts in state 1. Using the eigenvalues of $M$, find the probability that it is in state 1 after $5$ steps. Express your answer as a common fraction.