Block multiplication of matrices with a matrix having determinant 1

Question

Suppose we have two matrices, $A \in M_{m\times n} (\mathbb {F})$ and $B\in M_{m\times p} (\mathbb {F})$, where $C(B) \subseteq C(A)$. How do you show that there exists a matrix, $P \in M_{n+p\times n+p} (\mathbb {F})$ such that $|P|=1$ and that block multiplication, $\begin{bmatrix}A&B\end{bmatrix}P = \begin {bmatrix} A&0_{m,p}\end{bmatrix}$ is well-defined?