How to convert a $3\times 5$ matrix to a product of a $3\times 2$ and $2\times 5$ matrix? I have a matrix 
$$A = \begin{bmatrix}
 1 & 1 & 2 & 2 & 3\\
 1 & 0 & 1 & 2 & 2\\
 0 & 1 & 1 & 0 & 1\\
 \end{bmatrix},$$
 and I would like to write this as a product of a $3\times 2$ and $2\times 5$ matrix. 
By reducing the matrix to row echelon form I get the $2\times 5$ matrix $$\begin{bmatrix}
1 & 0 & 1 & 2 & 2\\
0 & 1 & 1 & 0 & 1
\end{bmatrix},$$
How do I get the $3\times 2$ matrix?
 A: Quoting from Rank factorization:

In practice, we can construct one specific rank factorization $A = CF$ as follows: we can compute $B$, the reduced row echelon form of $A$. Then 
  $C$ is obtained by removing from $A$ all non-pivot columns, and 
  $F$ by eliminating all zero rows of $B$.

In your case, $A$ has rank 2, and you already computed the reduced row echelon form without the zero rows as 
$$
F = \begin{bmatrix}
1 & 0 & 1 & 2 & 2\\
0 & 1 & 1 & 0 & 1
\end{bmatrix}
$$
$C$ is obtained by picking only the pivot-columns (one and two) from $A$
$$
C = \begin{bmatrix}
 1 & 1 \\
 1 & 0\\
 0 & 1\\
 \end{bmatrix}
$$
and $A=CF$ is the desired decomposition.
A: Each row of the $3×2$ matrix represents coefficients for a linear combination of the rows of the $2×5$ matrix that produces a row of the original matrix.
Using the found row-echelon form, let's consider the first row of the $3×2$ matrix:
$$(1,1,2,2,3)=a(1,0,1,2,2)+b(0,1,1,0,1)$$
We see that $a=b=1$, so the first row is $(1,1)$. Similar computations apply for the other two rows.
