Any hints on how one should determine the rank of a system of equations with parameters included? Solving it for different values of parameters, the usual way, as in an attempt to find the number of solutions of a system isn't the answer.
For instance, the following matrix:
$$\begin{pmatrix} k& 1& 1 \\ 1 &k &1\\ 1& 1& k\end{pmatrix}$$
For your example problem, I would try looking at the determinant of the matrix for the system of equations. So, in this case, we can calculate the determinant as $k^3-3k+2$. Factoring, we see this has two distinct roots, $k=1$ and $k=-2$. Now, if $k$ is not a root, then the determinant is non-zero and thus the system has full rank. We need to examine the $k=1,-2$ cases separately then: if $k=1$, all the columns are the same, and thus the system can only have rank 1. If $k=-2$, note that any two of the columns are not related by scalar multiplication, and hence independent. Thus, the system has rank 2.
In general, examining the determinant is often a useful way to approach these problems; it gives us a fairly immediate test for being full rank, and by finding where the determinant vanishes we can know where the system is below full rank. At this point, I'd say just plug those values in and try Gaussian elimination.