<s>[INST] 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 
$$C=\begin{bmatrix}
3/4 & 2/3 \\
3/4 & 2/3 \\
1/2 & 2/3 \end{bmatrix}$$
 [/INST] There are exactly $9$ matrices in $A$. The dimension of the vector space $V$ of $3\times 2$ matrices is $6$.
The rank of the matrix (formed putting each matrix of $A$ in a row)
$$\begin{pmatrix}
1&1&0&1&1&0\\
1&1&0&1&0&1\\
1&1&0&0&1&1\\
1&0&1&1&1&0\\
1&0&1&1&0&1\\
1&0&1&0&1&1\\
0&1&1&1&1&0\\
0&1&1&1&0&1\\
0&1&1&0&1&1
\end{pmatrix}$$
is $5$, and this means that $A$ does not generate $V$. The other answer shows that, nevertheless, the particular matrix $B$ is generated by $A$.
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