We study the asymptotic behavior of the smallest eigenvalue, lambda(N), of the Hankel (or moments) matrix denoted by HN=mu;m+n0 <= m,n <= N, with respect to the weight w(x)=x alpha e-x beta,x is an element of[0,infinity),alpha>-1,beta>12. An asymptotic expression of the polynomials orthogonal with w(x) is established. Using this, we obtain the specific asymptotic formulas of lambda(N) in this paper. Applying a parallel numerical algorithm, we get a variety of numerical results of lambda(N) corresponding to our theoretical calculations.