Answer
$P_{C\leftarrow B}=\begin{bmatrix}
8&3\\-5&-2
\end{bmatrix}$
$P_{B\leftarrow C}=\begin{bmatrix}
2&3\\-5&-8
\end{bmatrix}$
Work Step by Step
One way to find the change of coordinate matrix B to C is to row reduce the matrix
$\begin{bmatrix}
C&B\\
\end{bmatrix}$ to $\begin{bmatrix}
I&P_{C\leftarrow B}\\
\end{bmatrix}$
$\begin{bmatrix}
4&5&7&2\\1&2&-2&-1
\end{bmatrix}$~$\begin{bmatrix}
0&-3&15&6\\1&2&-2&-1
\end{bmatrix}$~$\begin{bmatrix}
1&2&-2&-1\\0&-3&15&6
\end{bmatrix}$~$\begin{bmatrix}
1&2&-2&-1\\0&1&-5&-2
\end{bmatrix}$~$\begin{bmatrix}
1&0&8&3\\0&1&-5&-2
\end{bmatrix}$
$P_{C\leftarrow B}=\begin{bmatrix}
8&3\\-5&-2
\end{bmatrix}$
We can calculate the inverse of the above matrix to find the change of coordinate matrix C to B.
$P_{B\leftarrow C}=\begin{bmatrix}
8&3\\-5&-2
\end{bmatrix}^{-1}=\begin{bmatrix}
2&3\\-5&-8
\end{bmatrix}$
