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6th Edit,So this strange...or is it? If I understand the procedure that we have worked out thus far, the plots that I created above tell me "how singular" my matrix is as a function of my parameter $\kappa$. Thus I would probably like my y-axis to be really small, and so I am telling SingularValueList to only give me the last entry since that should be the smallest singular value, and also why I'm using the tolerance function,so that the smallest value do not get ignored. One question is, why use tolerance if were are already looking at the smallest singular value? The other problem, the strange part, is that when I find a $\kappa$ using SVL, and root finding command, then write $\kappa = .508...$, and then Det[mat], I get something like 2.14^113, and 1.678^109, for another root. Is that right? Is this essentially as close to zero as we can get the determinant? Or am I missing something huge. Is it that I am using the wrong root. Is there one that can give me Det = .0000001 or even smaller? I guess its not all bad news, this smalleness of the Det could help me figure out which is the true root. Anyways just throwing some thoughts/questions out there. Thanks again to all who respond. This problem is starting to drive me crazy but I guess that research.

I was also thinking that maybe I could somehow get the matrix into an upper triangular form and then just multiply the diagonal elements. For this I tried using Mathematica's RowReduce command, but for some weird reason that just results in the identity matrix. I thought that RowReduce might give me an upper triangular matrix with $f(\kappa)$ on the diagonal, and I could just multiply the diagonal elements and get a polynomial for $\kappa$ and solve.

Which I looked up and is supposed to mean that root finder cannot find real roots. So my first question is: what does {x -> 0.506739}, mean if Mathematica couldn't find real roots?

I've also tried to increase the AccracyGoal and WorkingPresicion with this

So I'm quite lost as to where to go now. I've gone through my code and made sure that I put everything in fractional form, i.e. 1/2 instead of .5 since I know that can reduce precision, and make Mathematica angry.

Now there are also no errors when I try the findroot command.

I was also thinking that maybe I could somehow get the matrix into an upper triangular form and then just multiply the diagonal elements. For this I tried using Mathematica's RowReduce command, but for some weird reason that just results in the identity matrix. I thought that RowReduce might give me an upper triangular matrix with $f(\kappa)$ on the diagonal, and I could just multiply the diagonal elements and get a polynomial for $\kappa$ and solve.

Which I looked up and is supposed to mean that root finder cannot find real roots. So my first question is: what does {x -> 0.506739}, mean if Mathematica couldn't find real roots?

I've also tried to increase the AccracyGoal and WorkingPresicion with this

So I'm quite lost as to where to go now. I've gone through my code and made sure that I put everything in fractional form, i.e. 1/2 instead of .5 since I know that can reduce precision, and make Mathematica angry.

Now there are also no errors when I try the FindRoot command.

5th Edit New plot with Jens' suggestion used in code.

Now there are also no errors when I try the findroot command.