Suppose we have set of matrix equations, for this case lets consider a 2×2 set. The equations are:

\[Alpha]*x*840.92 == 470.44*(Coth[\[Alpha]] - 1/\[Alpha])+1329.74(Coth[[Beta]] - 1/\[Beta])  \[Beta]*x*840.92 == 664.87*(Coth[\[Alpha]] - 1/\[Alpha])+143.61(Coth[[Beta]] - 1/\[Beta]) 

I have tried using NSolve[] to no avail as it keeps the Coth[] unevaluated.

x=0.1 NSolve[{\[Alpha]*x*840.92 == 470.44*(Coth[\[Alpha]] - 1/\[Alpha]) + 1329.74*(Coth[\[Beta]] - 1/\[Beta]),\[Beta]*x *840.92 == 664.87*(Coth[\[Alpha]] - 1/\[Alpha]) + 143.61*(Coth[\[Beta]] - 1/\[Beta]) && \[Alpha] > 0 && \[Beta] >  0}, {\[Alpha], \[Beta]}, Reals] 

This just returns the same message.

The main idea is that for a given x, alpha and Beta can be calculated. I was trying to use a series expansion but the results are very reliable I think. This should be able to work for any size matrix.