Proving the 3-dimensional representation of S3 is reducible The 3-dimensional representation of the group S3 can be constructed by introducing a vector $(a,b,c)$ and permute its component by matrix multiplication.
For example, the representation for the operation $(23):(a,b,c)\rightarrow(a,c,b)$ is
$
D(23)=\left(\begin{matrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \end{matrix}\right)
$
and so forth. 
The exercise is to prove this representation is reducible. The hint tells me to find a common eigenvector for all 6 matrices which is just $(1,1,1)$. How do I proceed from here? Any help is appreciated.

You asked: The exercise is to prove this representation is
reducible. The hint tells me to find a common eigenvector for all
6 matrices which is just $(1,1,1)$. How do I proceed from here?
Any help is appreciated.
Representation is
irreducible
if and only if the corresponding $FG$-module has no non-trivial
$FG$-submodule. If you have managed to find a common eigenvector
$v$, then you have $vg=\lambda_g v$ for each $g\in G$; which
implies that $\operatorname{span}(v)$ is an $FG$-submodule.
In short: Looking for 1-dimensional submodules is the same thing
as looking for common eigenvectors. If your the whole $FG$-module
has dimension 3, it suffices to find out whether it has a
1-dimensional submodule, in order to decide whether it is
irreducible or not.
It is perhaps worth mentioning that the same approach would work for any $S_n$ and that this representation is called permutation representation of $S_n$.
Another interesting fact is that the permutation representation can be decomposed into this trivial representation and an irreducible representation of degree $n-1$. We have a question about this on this site; link to this MO thread is given there in comments.
Note: My answer is more-or-less the same as Benjalim's answer (which is deleted at the moment, so it is visible only for 10k+ users), with the exception that my answer uses modules and his answer avoids modules and uses only representations. (Both approaches, $FG$-modules and representations, are equivalent in the sense that we can get module from a representation and vice-versa. Hence we can describe properties of representation using the properties of the corresponding $FG$-module.)
