solving linear systems with parameters

I am trying to solve simple, small linear systems, with parameters. I would like to have the result shown for various values of the parameter. RowReduce[], Solve[], LinearSolve[], Reduce[], etc., all show "generic" solutions, without special values for the parameters. (Note that Wolfram|Alpha does show this extra analysis.)
If the matrix of coefficients is square, Det[] helps to find the parameter values of interest:

m = {{1, 1, k}, {1, k, 1}, {k, 1, 1}};
FindInstance[Det[m] == 0, k, Reals, 3]

produces: {{k -> -2}, {k -> 1}}.
However, I would like to work with non-square coefficient matrices as well.
I have tried MatrixRank[], e.g.:

FindInstance[MatrixRank[m] < 3, k, Reals, 3]

which just produces: {}. The same goes for all of the other functions which perform elimination (NullSpace[], RowReduce[], amongst others).
Any suggestions how to make this work?

Thank you

Editing: I was trying to keep it simple. For example:

FindInstance[MatrixRank[m] == 2, k, Reals, 3]

I would want the output of: {{k -> -2}} . I think if I can understand that, I will be able to handle the rest, and would be happy to include it here, once I have done that.

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