How to determine the values of integer matrices?

Question

If $A \times W=F$, where $A$ is $2 \times 2$ integer matrix, $W$ is $2 \times 4$ integer matrix and $F$ is $2 \times 4$ integer matrix.

If I know the value of $F$ ($2 \times 4$ integer matrix), i.e.,
\begin{bmatrix} 1 & 1 & 0 &1\\ 0 & -1 & -1 & 1 \end{bmatrix}

How can I determine the value of matrices $A$ and $W$?

Subject to:

1. In $W$ matrix, the values of elements $w13, w14, w21, w22$ = zero, and the values of $w11, w12, w23, w24 \neq 0$ must be integers and not equal zeros. i.e., \begin{bmatrix} w11 & w12 & 0 &0\\ 0 & 0 & w23 & w24 \end{bmatrix}

2. All elements of the $2 \times 2$ matrix $A$ are integers and the rank of $A$ must be 2.

Answer 1

With the given $F$, the equations to solve are $$ \eqalign{ a_{{1,1}}w_{{1,1}}&=1\cr a_{{1,1}}w_{{1,2}}&=1\cr a_{{1,2}}w_{{2,3}}&=0\cr a_{{1,2}}w_{{2,4}}&=1\cr a_{{2,1}}w_{{1,1}}&=0\cr a_{{2,1}}w_{{1,2}}&=-1\cr a_{{2,2}}w_{{2,3}}&=-1\cr a_{{2,2}}w_{{2,4}}&=1}$$ $w_{2,3} \ne 0$ from equation 7 and $a_{1,2} \ne 0$ from equation 4 so equation 3 is impossible.