I'm trying to find the determinant of a band diagonal matrix that has a parameter, $\kappa$, in some of the entries. Some entries are just numerical ones, others ($\kappa$ X number), while others are ($\kappa$ + number). I have been told that they way to solve for $\kappa$ is to find the determinant of the this matrix and then find values of $\kappa$ that make the determinant zero. 

 The main issue I'm having is that when my matrix becomes large the determinant just results to zero,and in other cases to calculation overflow. (I'm trying to work out all the bugs in the code, so det =0, might be some error I'm making, but the overflow error is not avoidable). 
 
 I have already tried an `LUDecomposition` on the matrix, and that seems to take forever, I don't have a problem waiting, but working out the scaling, it seemed like I would have to wait a couple of days for a 500X500 matrix, and my real problem might have to be done on a 1000X1000 matrix. 

 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.

Any and all help is greatly appreciated. I'm not really sure how to put up my code, or the matrix for that matter. That is the thing that would probably help you guys the most. If there is a way for me to put up the matrix please let me know. 

Thanks again. 

EDIT- A matrix that gives you guys some idea of my matrix. 

 t2 = {{-892.33, 973.21, 44.306 + \[Kappa], -81.103,0}, 
 {446.12, -557.94, 0, -682.54, -314.89}, 
 {0,893.37, -506.68*\ [Kappa],-391.457, 0}, {0, 429.78, 0, -210.47, 
 342.85}, {278.32*\[Kappa], 0, 963.41, 217.71, -342.68 + \[Kappa]}} 

**2nd-EDIT**
Although I do not fully understand what Jens' code is doing I did try it on my real matrix. The result is
 

 In[193]:= f[\[Kappa]_?NumericQ] := 
 Min[Diagonal[SingularValueDecomposition[mat][[2]]]]
 In[194]:= Plot[f[\[Kappa]], {\[Kappa], 0, 2}]

Well being a noob the site won't let me upload an image, but it basically looks like there should be roots around $\kappa$ = .1, .2, .4,.4, .6. 

So I tried to find the root using 

 In[196]:= FindRoot[f[x], {x, .5}]

And then I get a bunch of error messages. 

 During evaluation of In[196]:= InterpolatingFunction::dmval:
 Input value {-0.173686} lies outside the range of data 
 in the interpolating function.Extrapolation will be used. >>

 During evaluation of In[196]:= InterpolatingFunction::dmval: 
 Input value {-0.173686} lies outside the range of data in the 
 interpolating function. Extrapolation will be used. >>

 During evaluation of In[196]:= InterpolatingFunction::dmval: 
 Input value {-0.173686} lies outside the range of data 
 in the interpolating function. Extrapolation will be used. >>

 During evaluation of In[196]:= General::stop: Further output of 
 InterpolatingFunction::dmval will be suppressed during this calculation. >>

 Out[196]= {x -> -3.28829*10^-13}

So I figured that if root-finder couldn't do it, I'd just try it by hand, i.e. just look at the plot and keep narrowing down the point where f($\kappa$) =0, so I tried to evaluate

 In[190]:= f[.2]

which was taking forever considering that this command
 
 In[193]:= f[\[Kappa]_?NumericQ] := 
 Min[Diagonal[SingularValueDecomposition[mat][[2]]]]

and the plot command both took less than an second. I'm very confused. 

**3rd Edit**
I think I can post a picture now. So I will include my plot for `f[x]`. This might make it easier to figure out what is going wrong with root-finder. I'm thinking its the multiple roots. 

![enter image description here][1]


 [1]: https://i.sstatic.net/bDhDX.png