Decomposing a matrix into binary ones Let $A$ denote the set of all 3×2 binary matrices (those containing only 0's and 1's) in which the sum of each column adds up to 2. Can I decompose
$$B=\begin{bmatrix}
3/4 & 1/2 \\
3/4 & 3/4 \\
1/2 & 3/4 \end{bmatrix}$$
into a linear combination of matrices in $A$? For example,
$$B=1/2 \begin{bmatrix}1 & 1 \\ 0 & 0 \\ 1 & 1\end{bmatrix} + 1/2 \begin{bmatrix}1 & 0 \\ 0 & 1 \\ 1 & 1\end{bmatrix}$$
or something like that.
EDIT: I want to show there is at least a matrix that cannot be decomposed. How about 
$$C=\begin{bmatrix}
3/4 & 2/3 \\
3/4 & 2/3 \\
1/2 & 2/3 \end{bmatrix}$$

Multiply $B$ by four:
$$4B=\begin{bmatrix}3&2\\3&3\\2&3\end{bmatrix}$$
Finding a decomposition into matrices in $A$ is then easy:
$$4B=\begin{bmatrix}1&1\\0&0\\1&1\end{bmatrix}+\begin{bmatrix}0&1\\1&1\\1&0\end{bmatrix}+2\begin{bmatrix}1&0\\1&1\\0&1\end{bmatrix}$$
$$B=\frac14\begin{bmatrix}1&1\\0&0\\1&1\end{bmatrix}+\frac14\begin{bmatrix}0&1\\1&1\\1&0\end{bmatrix}+\frac12\begin{bmatrix}1&0\\1&1\\0&1\end{bmatrix}$$
