numeric computation of a partly decomposable matrix

Question

In combinatorics (van Lint, Wilson,A course in Combinatorics, 2nd ed., p.111), we learn that a partly decompsable matrix $A$ multiplied by two permutation matrices $P$ and $Q$ from each side has the form $$ P~A~Q = \begin{bmatrix}B&C\\ 0&D\end{bmatrix} $$ where $B$, $C$, $D$ are square matrices, and $0$ is the all-null matrix. To get started, let me try a simple numerical example.

1. How do $P$,$A$,$Q$ look like if $B$ is an all-two, $C$ is an all-three and $D$ is an all-four matrix each having dimension $2x2$? [Of course the example can be modified, so that $P$,$A$ and $Q$ can be computed]
2. When $C=0$, what are the forms of $P$, $A$, $Q$ now?

Sorry, I am a beginner in combinatorics.

Answer 1

In the meantime, I found out that a LU factorization with full pivoting can be used to solve this problem.

Matlab-code [L,U,P,Q]=lu(S) where S=sparse(A).