Answer
$P_{B \leftarrow C}=\begin{bmatrix}
\frac{1}{2} & -\frac{1}{2} & 0 & 0 \\
-\frac{7}{6} & \frac{5}{6} & \frac{1}{3} & 0 \\
-\frac{11}{30} & \frac{13}{30} & \frac{2}{15} & -\frac{1}{5} \\
\frac{1}{2} & 0 & 0 & 0
\end{bmatrix}$
Work Step by Step
From problem 20 we have $P_{C \leftarrow B} =\begin{bmatrix}
0 & 0 & 0 & 2\\
-2 & 0 & 0 & 2\\
5 & 3 & 0 & 2 \\
-1 & 2 & -5 & 2
\end{bmatrix}$
Since $P_{B \leftarrow C}=(P_{C \leftarrow B})^{-1}$ we have $P_{B \leftarrow C}=\begin{bmatrix}
\frac{1}{2} & -\frac{1}{2} & 0 & 0 \\
-\frac{7}{6} & \frac{5}{6} & \frac{1}{3} & 0 \\
-\frac{11}{30} & \frac{13}{30} & \frac{2}{15} & -\frac{1}{5} \\
\frac{1}{2} & 0 & 0 & 0
\end{bmatrix}$