Answer
$P=\begin{bmatrix}
\frac{1}{4} & \frac{3}{4} \\
\frac{1}{2} & \frac{1}{2} \\
\end{bmatrix}$
Work Step by Step
Write the $ij$th entry of $P$ as the probability of transition from the state $i$ to $j$:
$1→1$: $\frac{1}{4}$
$1→2$: $\frac{3}{4}$
$2→1$: $\frac{1}{2}$
$2→2$: $\frac{1}{2}$
So:
$P=\begin{bmatrix}
\frac{1}{4} & \frac{3}{4} \\
\frac{1}{2} & \frac{1}{2} \\
\end{bmatrix}$
