Answer
$P_{B \leftarrow C}=\begin{bmatrix}
\frac{1}{3} & -\frac{2}{27} & -\frac{7}{27}\\
-\frac{1}{3} & \frac{14}{27} & \frac{22}{27} \\
0 & \frac{1}{9} & -\frac{1}{9}
\end{bmatrix}$
Work Step by Step
From problem 19 we have $P_{C \leftarrow B} =\begin{bmatrix}
4 & 1 & -2\\
1 & 1 & 5\\
1 & 1 & -4
\end{bmatrix}$
Since $P_{B \leftarrow C}=(P_{C \leftarrow B})^{-1}$ we have $P_{B \leftarrow C}=\begin{bmatrix}
\frac{1}{3} & -\frac{2}{27} & -\frac{7}{27}\\
-\frac{1}{3} & \frac{14}{27} & \frac{22}{27} \\
0 & \frac{1}{9} & -\frac{1}{9}
\end{bmatrix}$
