= Solution
The <matrix multiplication> $A\mathbf c$ adds each row. Equating it to $\mathbf b$ gives the <linear system>
$$
\alpha+\beta+\gamma=2,\qquad\alpha+3\beta+\gamma=4,\qquad2\alpha+4\beta+2\gamma=6.
$$
The last equation is the sum of the first two, so it supplies no further restriction. Subtracting the first from the second gives $2\beta=2$, hence $\beta=1$, and then $\gamma=1-\alpha$. Thus all parameter choices are
$$
\boxed{\alpha\in\mathbb R,\qquad\beta=1,\qquad\gamma=1-\alpha.}
$$
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