I trying to solve the following matrix

A:=Matrix([[alpha_O-E, beta_CO, 0, 0],[beta_CO, alpha_C-E, beta_CC, 0],[0, beta_CC, alpha_C-E, beta_CC],[0, 0, beta_CC, alpha_C-E]]);

by setting the determinant to zero and finding the roots in terms of parameters alpha_O, beta_CO, alpha_C, and beta_CC.  I have tried to use solve which returns

I several peaks from emission spectra which I am trying to fit wit a Gaussian distribution. 

y = y_0 + A*exp[-{(x-x_0)/(width)}^2

The problem I am running into is I am getting fits which are wrong.  Maple returns:

fit_658 := 1.+1.*exp(.50000000000000*(z-1.)^2)

The correct answer is:

-7e05+2.7e06*exp(((z-656.1)/9)^2)

Also, I am interested in finding the deviation in the parameters which are found.

I have two sets of scatter data.  Each set has been fit by a different function which is dependent on independent variable N.  The x-axis is 1/N. 

Set 1:

Scatter data is (0.125, -4.72557), (0.08333, -4.53944), (0.0625, -4.48229), (0.05, -4.42733)

where the x-axis is 1/N with N = 8, 12, 16, 20 and the y-axis is a function in terms of N as well: -8.55704612297018*SQRT(1-0.73962210555774*COS(PI()/(N+1)))

Set 2:

Scatter data is (0.125, -1.7312), (0.08333, -1.96658), (0.0625, -2.1274), (0.05, -2.187)

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I am trying to find parameters using NonlinearFit and keep encountering the following message:

'Warning, limiting number of iterations reached'

I have not been able to figure out how to overcome this warning - the fits I am getting are
 not very good.