I'm trying to find the determinant of a band diagonal matrix that has a parameter, κ, in some of the entries. Some entries are just numerical ones, others (κ X number), while others are (κ + number). I have been told that they way to solve for κ is to find the determinant of the this matrix and then find values of κ 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 multiple 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 row reduce might give me an upper triangular matrix with f(κ) on the diagonal , and I could just multiple the diagonal elements and get a polynomial for κ 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.