  We give an upper bound on the total variation distance between the linear
eigenvalue statistic, properly scaled and centred, of a random matrix with a
variance profile and the standard Gaussian random variable. The second order
Poincar\'e inequality type result is used to establish the bound. Using this
bound we prove Central limit theorem for linear eigenvalue statistics of random
matrices with different kind of variance profiles. We re-establish some
existing results on fluctuations of linear eigenvalue statistics of some well
known random matrix ensembles by choosing appropriate variance profiles.
